DISCLAIMER: This answer comes from a viewpoint that communion is a picture only, a memorial of Christ's work, without any efficacious aspect to it for salvation. That view is too much to work to defend fully in this answer, but some aspects of it will be apparent.
The Meaning of the Passage in John
The passage in John 6:53-58 is not a communion passage (though it is setting up the picture of what communion will celebrate). Rather, that passage needs to be understood in light of the prior verses, in which v.26 is a reasonable start point (all Scripture from NKJV; bold added for further discussion):
26 Jesus answered them and said, “Most assuredly, I say to you, you seek Me, not because you saw the signs, but because you ate of the loaves and were filled. 27 Do not labor for the food which perishes, but for the food which endures to everlasting life, which the Son of Man will give you, because God the Father has set His seal on Him.”
28 Then they said to Him, “What shall we do, that we may work the works of God?”
29 Jesus answered and said to them, “This is the work of God, that you believe in Him whom He sent.”
In these verses, Jesus begins to set up the idea that a person's "work" ought not be for earthly food, but some type of food that "endures to everlasting life." They ask, what is it they need to work (labor) then for the work of God, to which Jesus replies "believe in Him whom [God the Father] sent." Then the passage continues:
30 Therefore they said to Him, “What sign will You perform then, that we may see it and believe You? What work will You do? 31 Our fathers ate the manna in the desert; as it is written, ‘He gave them bread from heaven to eat.’ ”
32 Then Jesus said to them, “Most assuredly, I say to you, Moses did not give you the bread from heaven, but My Father gives you the true bread from heaven. 33 For the bread of God is He who comes down from heaven and gives life to the world.”
34 Then they said to Him, “Lord, give us this bread always.”
35 And Jesus said to them, “I am the bread of life. He who comes to Me shall never hunger, and he who believes in Me shall never thirst. 36 But I said to you that you have seen Me and yet do not believe. 37 All that the Father gives Me will come to Me, and the one who comes to Me I will by no means cast out. 38 For I have come down from heaven, not to do My own will, but the will of Him who sent Me. 39 This is the will of the Father who sent Me, that of all He has given Me I should lose nothing, but should raise it up at the last day. 40 And this is the will of Him who sent Me, that everyone who sees the Son and believes in Him may have everlasting life; and I will raise him up at the last day.”
In this next section, they seek to know what Christ's "work" will be that they may believe. The reply is the He is the bread from God and in essence, His work is to feed those who come to him in faith (v.35), which is doing the Father's will (v.38), and it is what brings everlasting life (v.40).
41 The Jews then complained about Him, because He said, “I am the bread which came down from heaven.” 42 And they said, “Is not this Jesus, the son of Joseph, whose father and mother we know? How is it then that He says, ‘I have come down from heaven’?”
43 Jesus therefore answered and said to them, “Do not murmur among yourselves. 44 No one can come to Me unless the Father who sent Me draws him; and I will raise him up at the last day. 45 It is written in the prophets, ‘And they shall all be taught by God.’ Therefore everyone who has heard and learned from the Father comes to Me. 46 Not that anyone has seen the Father, except He who is from God; He has seen the Father. 47 Most assuredly, I say to you, he who believes in Me has everlasting life. 48 I am the bread of life. 49 Your fathers ate the manna in the wilderness, and are dead. 50 This is the bread which comes down from heaven, that one may eat of it and not die. 51 I am the living bread which came down from heaven. If anyone eats of this bread, he will live forever; and the bread that I shall give is My flesh, which I shall give for the life of the world.”
52 The Jews therefore quarreled among themselves, saying, “How can this Man give us His flesh to eat?”
The Jews are taken aback by Jesus's statement of being the bread and having come from heaven (v.41), but Jesus reiterates that belief is central to everlasting life (v.47), and He as the "bread of life" (v.48), the "bread ... from heaven" (v.50, 51), "the living bread" (v.51) involves that bread, which "is My flesh," being given "for the life of the world" (this is the "work" that Christ does in "giving" of Himself; that is, His coming crucifixion). So when He says in v.51 "if anyone eats of this bread, he will live forever," He has already made the parallel that "eating" is symbolic of partaking of the bread, partaking of His flesh, i.e. believing in Him and His work. So the parallel picture set up is:
believe in Him --> everlasting life
eating of His flesh --> live forever
Eating symbolizes belief in the passage; the willingness to partake of Who Christ is and what He has done. The Jews did not get this, and question what Jesus means (v.52). The explanation is given in the OPs original passage, vv.53-58, which says:
53 Then Jesus said to them, “Most assuredly, I say to you, unless you eat the flesh of the Son of Man and drink His blood, you have no life in you. 54 Whoever eats My flesh and drinks My blood has eternal life, and I will raise him up at the last day. 55 For My flesh is food indeed, and My blood is drink indeed. 56 He who eats My flesh and drinks My blood abides in Me, and I in him. 57 As the living Father sent Me, and I live because of the Father, so he who feeds on Me will live because of Me. 58 This is the bread which came down from heaven—not as your fathers ate the manna, and are dead. He who eats this bread will live forever.”
Jesus expands the point purely in the symbolism. He is purposefully keeping the Jews in the dark about the meaning if they could not grasp it before (much as he does with his parables; though the answer was right before them in the eating = believing parallel). He adds a second part that it is not just a partaking of His flesh, but also of His blood that is needed (vv.53-56, though blood can be contained within the initial idea of "flesh" in the prior verses, and in the final wrap up in v.58, I think there are reasons not made apparent here why he split it out here as flesh and blood, but in part it is going to "mesh" with the communion picture later instituted).
So "feeding" on Christ (v.57) is a picture of partaking in Who Christ is and what He has done (sacrifice His flesh, shed His blood).
The Communion Passage Parallels
So then the communion passages that speak to partaking of the bread and wine (which, as the OP correctly notes, happened during the Passover celebration) set up the picture that the earthly bread represents the flesh of Christ and the earthly wine (itself represented by the "cup") the blood of Christ:
Luke 22:19 And He took bread, gave thanks and broke it, and gave it to them, saying, “This is My body which is given for you; do this in remembrance of Me.”
20 Likewise He also took the cup after supper, saying, “This cup is the new covenant in My blood, which is shed for you.
Matthew 26:26 And as they were eating, Jesus took bread, blessed and broke it, and gave it to the disciples and said, “Take, eat; this is My body.”
27 Then He took the cup, and gave thanks, and gave it to them, saying, “Drink from it, all of you. 28 For this is My blood of the new covenant, which is shed for many for the remission of sins. 29 But I say to you, I will not drink of this fruit of the vine from now on until that day when I drink it new with you in My Father’s kingdom.”
Mk 14:22 And as they were eating, Jesus took bread, blessed and broke it, and gave it to them and said, “Take, eat; this is My body.”
23 Then He took the cup, and when He had given thanks He gave it to them, and they all drank from it. 24 And He said to them, “This is My blood of the new covenant, which is shed for many. 25 Assuredly, I say to you, I will no longer drink of the fruit of the vine until that day when I drink it new in the kingdom of God.”
Luke's text, v.19, states this ritual of communion was specifically for "remembrance" purposes, a memorial of what Christ was about to do on the cross. But these passages are "reversing" the symbolism of the John 6 passage. In John 6, Jesus was using the idea of actual bread and drink to symbolize the reality of partaking by faith in His flesh and blood sacrifice; in the communion passages, He is taking the reality of bread and wine to symbolize the believers partaking by their faith in the idea of what Christ has actually done through His flesh and blood.
So in 1 Cor 11:23-34, the symbolism is still being carried on in churches, but a warning is given:
23 For I received from the Lord that which I also delivered to you: that the Lord Jesus on the same night in which He was betrayed took bread; 24 and when He had given thanks, He broke it and said, “Take, eat; this is My body which is broken for you; do this in remembrance of Me.” 25 In the same manner He also took the cup after supper, saying, “This cup is the new covenant in My blood. This do, as often as you drink it, in remembrance of Me.”
26 For as often as you eat this bread and drink this cup, you proclaim the Lord’s death till He comes.
Notice the emphasis still on remembrance and that this ritual is a way for those participating to "proclaim the Lord's death" until His second coming. Then the warning:
27 Therefore whoever eats this bread or drinks this cup of the Lord in an unworthy manner will be guilty of the body and blood of the Lord. 28 But let a man examine himself, and so let him eat of the bread and drink of the cup. 29 For he who eats and drinks in an unworthy manner eats and drinks judgment to himself, not discerning the Lord’s body. 30 For this reason many are weak and sick among you, and many sleep. 31 For if we would judge ourselves, we would not be judged. 32 But when we are judged, we are chastened by the Lord, that we may not be condemned with the world.
33 Therefore, my brethren, when you come together to eat, wait for one another. 34 But if anyone is hungry, let him eat at home, lest you come together for judgment. And the rest I will set in order when I come.
Eating in an "unworthy manner" is tied to "not discerning the Lord's body." In other words, not believing that Christ actually was the bread of life and actually did the sacrificial work on the cross that He said He did for them. They are partaking in the earthly symbol (per the Gospels institution of communion) without having partaken (believed) in the reality the symbol points to (per the John passage). In so doing, one is self-deceptively celebrating "judgment to himself" (v.29; they are already under judgement by unbelief [Mk 16:16, Jn 3:18]; here they are inadvertently celebrating that), rather than celebrating Christ (vv.24-26), so a person needs to "examine himself" (v.28) to see where their faith really resides and why they are partaking of this communion memorial; to fail to do so because of their faith in Christ makes them "guilty" (v.27) of the additional sin of partaking in the communion for all the wrong reasons. And yes, the passage indicates that this additional guilt can manifest in such unbelievers among them being weak, sick, and even dying for doing so (v.30; I believe stages may occur, as God gives grace for even such a person to come to faith and join rightly into the communion, so these who are judged, who did not judge themselves [v.31], experience a time of chastening in hopes that they come to faith and so avoid condemnation with the rest of the world). So the final exhortation is to not partake of the communion for purposes of just getting some food (v.33-34), as that is clearly the wrong focus for this memorial service.
The Passover Picture
Briefly, it is worth noting that the Passover meal was also a picture memorializing the salvation God made for the people of Israel at the time of the Exodus by the death of a lamb and the application of the blood over the house to keep those who partook of that sacrifice from being judged as God passed over Egypt and brought physical death to the firstborn. So the parallel to communion is clear in that it memorializes the salvation God made for all people through by the death of the Lamb (Jn 1:29, 1 Cor 5:7, 1 Pet 1:19, Rev 5:6 et al.) and the application of His blood to keep those partaking in His sacrifice from being judged as God condemns the unbelieving during the second death (eternal death), the casting into the lake of fire (Rev 20:14-15).
Partaking or not of communion has no bearing on a believer's having everlasting life. They may partake "as often" as they choose (1 Cor 11:25-26); however, it is expected under normal circumstances to partake at times, for Christ does command to "do this in remembrance" (Lk 22:19); but there is no judgment associated with not following through on that. Per the points above, Paul in 1 Corinthians 11 gives the clearest testimony that the 1st Century church did not consider communion a requirement for eternal life, but a memorial of that life gained in Christ.
But partaking of it at any time as an unbeliever can hasten their passing in life from this earth and hasten their coming to judgment if they do not come to faith before it is too late.