The LXX reads thus:
Micah 2:12-13 (LXX) συναγόμενος συναχθήσεται Ιακωβ σὺν πᾶσιν ἐκδεχόμενος ἐκδέξομαι τοὺς καταλοίπους τοῦ Ισραηλ ἐπὶ τὸ αὐτὸ θήσομαι τὴν ἀποστροφὴν αὐτῶν ὡς πρόβατα ἐν θλίψει ὡς ποίμνιον ἐν μέσῳ κοίτης αὐτῶν ἐξαλοῦνται ἐξ ἀνθρώπων · διὰ τῆς διακοπῆς πρὸ προσώπου αὐτῶν διέκοψαν καὶ διῆλθον πύλην καὶ ἐξῆλθον δ αὐτῆς καὶ ἐξῆλθεν ὁ βασιλεὺς αὐτῶν πρὸ προσώπου αὐτῶν ὁ δὲ κύριος ἡγήσεται αὐτῶν
Brenton translates thus:
Micah 2:12-13 (Brenton) Jacob shall be completely gathered with all [his people]: I will surely receive the remnant of Israel; I will cause them to return together, as sheep in trouble, as a flock in the midst of their fold: they shall rush forth from among men through the breach made before them: they have broken through, and passed the gate, and gone out by it: and their king has gone out before them, and the Lord shall lead them.
The Hebrew reads as follows (WLC):
אָסֹף אֶאֱסֹף יַעֲקֹב כֻּלָּךְ קַבֵּץ אֲקַבֵּץ שְׁאֵרִית יִשְׂרָאֵל יַחַד אֲשִׂימֶנּוּ כְּצֹאן בָּצְרָה כְּעֵדֶר בְּתֹוךְ הַדָּֽבְרֹו תְּהִימֶנָה מֵאָדָֽם׃ 12
13 עָלָה הַפֹּרֵץ לִפְנֵיהֶם פָּֽרְצוּ וַֽיַּעֲבֹרוּ שַׁעַר וַיֵּצְאוּ בֹו וַיַּעֲבֹר מַלְכָּם לִפְנֵיהֶם וַיהוָה בְּרֹאשָֽׁם׃
Jerome translates the Hebrew thus (English via the Douay-Rheims translation of the Vulgate):
Micah 2:12-13 (DRB) I will assemble and gather together all of thee, O Jacob: I will bring together the remnant of Israel, I will put them together as a flock in the fold, as the sheep in the midst of the sheepcotes, they shall make a tumult by reason of the multitude of men. 13 For he shall go up that shall open the way before them: they shall divide, and pass through the gate, and shall come in by it: and their king shall pass before them, and the Lord at the head of them.
(I think this is a much better translation of the Hebrew than the rather awkward translation of the KJV.)
The Greek of John 10:1-16 reads as follows (NA28):
John 10:1-16 (NA28) Ἀμὴν ἀμὴν λέγω ὑμῖν, ὁ μὴ εἰσερχόμενος διὰ τῆς θύρας εἰς τὴν αὐλὴν τῶν προβάτων ἀλλ’ ἀναβαίνων ἀλλαχόθεν ἐκεῖνος κλέπτης ἐστὶν καὶ λῃστής· ὁ δὲ εἰσερχόμενος διὰ τῆς θύρας ποιμήν ἐστιν τῶν προβάτων. τούτῳ ὁ θυρωρὸς ἀνοίγει καὶ τὰ πρόβατα τῆς φωνῆς αὐτοῦ ἀκούει καὶ τὰ ἴδια πρόβατα φωνεῖ κατ’ ὄνομα καὶ ἐξάγει αὐτά. ὅταν τὰ ἴδια πάντα ἐκβάλῃ, ἔμπροσθεν αὐτῶν πορεύεται καὶ τὰ πρόβατα αὐτῷ ἀκολουθεῖ, ὅτι οἴδασιν τὴν φωνὴν αὐτοῦ· ἀλλοτρίῳ δὲ οὐ μὴ ἀκολουθήσουσιν, ἀλλὰ φεύξονται ἀπ’ αὐτοῦ, ὅτι οὐκ οἴδασιν τῶν ἀλλοτρίων τὴν φωνήν. Ταύτην τὴν παροιμίαν εἶπεν αὐτοῖς ὁ Ἰησοῦς, ἐκεῖνοι δὲ οὐκ ἔγνωσαν τίνα ἦν ἃ ἐλάλει αὐτοῖς. Εἶπεν οὖν πάλιν ὁ Ἰησοῦς· ἀμὴν ἀμὴν λέγω ὑμῖν ὅτι ἐγώ εἰμι ἡ θύρα τῶν προβάτων. πάντες ὅσοι ἦλθον [πρὸ ἐμοῦ] κλέπται εἰσὶν καὶ λῃσταί, ἀλλ’ οὐκ ἤκουσαν αὐτῶν τὰ πρόβατα. ἐγώ εἰμι ἡ θύρα· δι’ ἐμοῦ ἐάν τις εἰσέλθῃ σωθήσεται καὶ εἰσελεύσεται καὶ ἐξελεύσεται καὶ νομὴν εὑρήσει. ὁ κλέπτης οὐκ ἔρχεται εἰ μὴ ἵνα κλέψῃ καὶ θύσῃ καὶ ἀπολέσῃ· ἐγὼ ἦλθον ἵνα ζωὴν ἔχωσιν καὶ περισσὸν ἔχωσιν. Ἐγώ εἰμι ὁ ποιμὴν ὁ καλός. ὁ ποιμὴν ὁ καλὸς τὴν ψυχὴν αὐτοῦ τίθησιν ὑπὲρ τῶν προβάτων· ὁ μισθωτὸς καὶ οὐκ ὢν ποιμήν, οὗ οὐκ ἔστιν τὰ πρόβατα ἴδια, θεωρεῖ τὸν λύκον ἐρχόμενον καὶ ἀφίησιν τὰ πρόβατα καὶ φεύγει– καὶ ὁ λύκος ἁρπάζει αὐτὰ καὶ σκορπίζει–ὅτι μισθωτός ἐστιν καὶ οὐ μέλει αὐτῷ περὶ τῶν προβάτων. Ἐγώ εἰμι ὁ ποιμὴν ὁ καλὸς καὶ γινώσκω τὰ ἐμὰ καὶ γινώσκουσίν με τὰ ἐμά, καθὼς γινώσκει με ὁ πατὴρ κἀγὼ γινώσκω τὸν πατέρα, καὶ τὴν ψυχήν μου τίθημι ὑπὲρ τῶν προβάτων. καὶ ἄλλα πρόβατα ἔχω ἃ οὐκ ἔστιν ἐκ τῆς αὐλῆς ταύτης· κἀκεῖνα δεῖ με ἀγαγεῖν καὶ τῆς φωνῆς μου ἀκούσουσιν, καὶ γενήσονται μία ποίμνη, εἷς ποιμήν.
Again, Jerome translates it thus:
John 10:1-16 (DRB) Amen, amen I say to you: He that entereth not by the door into the sheepfold, but climbeth up another way, the same is a thief and a robber. 2 But he that entereth in by the door is the shepherd of the sheep. 3 To him the porter openeth; and the sheep hear his voice: and he calleth his own sheep by name, and leadeth them out. 4 And when he hath let out his own sheep, he goeth before them: and the sheep follow him, because they know his voice. 5 But a stranger they follow not, but fly from him, because they know not the voice of strangers. 6 This proverb Jesus spoke to them. But they understood not what he spoke to them. 7 Jesus therefore said to them again: Amen, amen I say to you, I am the door of the sheep. 8 All others, as many as have come, are thieves and robbers: and the sheep heard them not. 9 I am the door. By me, if any man enter in, he shall be saved: and he shall go in, and go out, and shall find pastures. 10 The thief cometh not, but for to steal, and to kill, and to destroy. I am come that they may have life, and may have it more abundantly. 11 I am the good shepherd. The good shepherd giveth his life for his sheep. 12 But the hireling, and he that is not the shepherd, whose own the sheep are not, seeth the wolf coming, and leaveth the sheep, and flieth: and the wolf catcheth, and scattereth the sheep: 13 And the hireling flieth, because he is a hireling: and he hath no care for the sheep. 14 I am the good shepherd; and I know mine, and mine know me. 15 As the Father knoweth me, and I know the Father: and I lay down my life for my sheep. 16 And other sheep I have, that are not of this fold: them also I must bring, and they shall hear my voice, and there shall be one fold and one shepherd.
The only real difference here is that Jesus uses the term 'door' instaed of 'gate.' This can be explained by the fact that this is a parable, where Jesus often blends the specific with the more general (or the specific with relation to the parable, to the general with respect to its doctrinal application). Also, it could just be another way to render the Hebrew שַׁעַר (gate), since it is quite generic and just means 'portal' or 'entrance.'
Is it possible that Jesus was unaware of this Scripture when He identifies Hismelf as the Gate here? Moreover, Mt 7:14, and Lk 13:24 absolutely conflate the two Greek words.
It's almost certain Jesus used the Aramaic equivalent of the Hebrew שַׁעַר meaning gate, which John and Matthew translate θύρας, and Luke πύλη. I think it's that simple, and that this is indeed a rather direct allusion to this passage. The correspondance are two many, and the topic too similar.