With that said, I do think there is a clear allusion here to the tabernacling of God with his people (and σκηνόω is the same verb used in Ex. 33:7ff LXX), and there is almost certainly wordplay with these Greek and Hebrew consonantal roots.
ἐσκήνωσεν. The verb σκηνοῦν occurs four times in the NT, here, and in Rev 7:15; 18:6; 21:3. Only Rev 21:3 is akin to our v. 14: Ἰδοὺ ἡ σκηνὴ τοῦ θεοῦ μετὰ τῶν ἀνθρώπων, καὶ σκηνώσει μετʼ αὐτῶν. Nor, on the face of it, is κατασκηνοῦν of much help. Three of its four NT occurrences (Mt 13:32 = Mk 4:32 = Lk 13:19) refer to birds nesting, as do both instances of the cognate κατασκήνωσις (Mt 8:20 = Lk 9:58). The one other text (Acts 2:26) is a quotation from Ps 16:9 LXX (ἡ σάρξ μου κατασκηνώσει ἐπʼ ἐλπίδι).
The LXX contains only five instances of σκηνοῦν: Gen 13:12; Judg 5:17 (in B [×2]): 8:11; 3 Kgdms 8:12; all except the last refer to those who lived, physically, in material tents. This is certainly not the sense (even metaphorically) of ἐσκήνωσεν in 1:14. By contrast, κατασκηνόω (rare in Classical Greek: see LSJ) is found in the LXX (including variant readings) more than 60 times, in 55 of which it translates the Hebrew root שׁכן (shakan). If one asks why, as a translation of this Hebrew verb, the Greek compound form is so much more frequent than the simplex, it is probably because the prefix κατα—brings out the idea of a long, and even permanent, residence: it is regularly used of Israel’s dwelling in the Promised Land: e.g. Num 14:30; Deut 33:12, 28; Josh 22:29; 2 Kgdms 7:10 (cf. κατοικεῖν, καταπαύειν etc.). κατασκηνοῦν is also used for God’s first coming down to dwell in the Land (Num 35:34), or to dwell in the Temple (3 Kgdms 6:13; 1 Chr 23:25; 2 Chr 6:1); and after the destruction of the Temple, there is the promise that ‘his name’ would again dwell there (Ezek 43:7 and Neh 1:9; cf. Jer 7:12, of Shiloh). Thus κατεσκήνωσεν would have suited the theme admirably, but John does not use that form, and it is hard to think it was because he wished to stress that Jesus’ life on earth was so transitory.
In fact, to understand the true thrust of John’s ἐσκήνωσεν, we must turn from the LXX to the Hebrew, and some introductory remarks are required. A few altars were built here and there on special occasions by the patriarchs, but they were not places of continuing worship; the Israelites had no permanent sanctuary until after the Covenant at Sinai (Exod 24). When Yahweh had revealed his proper name at Exod 3:15, his first command, immediately after the sealing of the Covenant, was to gather materials to make a great tent, to be the visible sign of his presence in the camp (Exod 25–31). Two verses at the beginning of this passage are then of the utmost importance. (a) Exod 25:8. This is the first occasion in the Bible that Yahweh says, in a phrase that was to be central to his covenant-promises, וְשָׁכַנְתִּי בְּתוֹכָם (wĕšākanti bĕtokām), ‘and I shall dwell among you’. The same, or an almost identical, Hebrew phrase recurs, a second time, at Exod 29:45 (a promise, to be fulfilled after the consecration of the Tent); thirdly, at 1 Kgs 6:13, at the Dedication of the Temple; fourthly, at Ezek 43:7, and Zech 2:10, 11 RSV [Heb 2:14, 15], with reference to the divine presence returning to the Temple after the Babylonian Exile. These are the only six instances of the word וְשָׁכַנְתִּי (wĕšākanti = ‘and I shall dwell’) in the Hebrew Bible, and it is noteworthy that it is uttered only by Yahweh, the all-merciful, i.e. the speaker is always called ‘Yahweh’, never ‘God’. (b) In Exod 25:9 we encounter the noun הַמִּשְׁכָּן (hammiškan: LXX σκηνή), this too at its first time in the Hebrew Bible. It is commonly translated into English as the tabernacle or the Tent, but means literally the dwelling, the abode. A cognate of shakan, it occurs 139 times in the MT, and with two or three exceptions, wherever it is found in the singular, it denotes always (= 136 times) the ‘physical abode’ of Yahweh (BDB). This verb and this noun, each occurring, with reference to God’s indwelling in the midst of Israel, for the first time in Exod 25, explain why John chose to write ἐσκήνωσεν; and they, by recalling Exod 25, and the dedication of Solomon’s Temple, and the return after the Exile, disclose the full meaning of ἐσκήνωσεν. A medieval Latinist caught the OT allusions perfectly: et tabernaculavit in nobis.
Further, at 25:8, Aquila, Symmachus and Theodotion all render the verb וְשָׁכַנְתִּי (wĕšākanti) normally, and quite correctly, as σκηνώσω. Aquila was much given to translating a Hebrew root by a similar Greek word, and the correspondence between shakan and σκηνοῦν must have seemed too good to miss. Perhaps it was noticed by Theodotion and Symmachus too; this would explain why three such disparate versions could, in Exod 25:8, arrive at the rarely used simplex σκηνοῦν instead of the commonly used compound κατασκηνοῦν. May not John have done the same?
Thus the καί before ἐσκήνωσεν marks yet another climax. When the Logos, the Memra, became flesh, there was the final dwelling-place of God among men, as Rev 21:3 declares: Ἰδοὺ ἡ σκηνὴ τοῦ θεοῦ μετὰ τῶν ἀνθρώπων, καὶ σκηνώσει μετʼ αὐτῶν, where σκηνώσει must refer to God, especially as the verse continues with ‘and God himself shall be with them’. Revelation 21:22 completes this picture of new Jerusalem with ‘and I saw no temple in her, for the Lord God the Almighty is her temple, and the Lamb’. As in Rev 21–22, so throughout John’s Gospel, there runs the theme of ‘the Temple of his body’: see 2:21; 19:34, 37; 20:25, 27.
ἐν ἡμῖν refers primarily to those who personally ‘heard, saw and touched’ Jesus (1 Jn 1:1), secondly to that particular generation, whether Jewish or Gentile; thirdly and by extension to all who hear the Gospel story to the end of time; and to all the human race.
Before moving to 14c, it is worth asking why the LXX did not always translate the verb וְשָׁכַנְתִּי (wĕšākanti) by καὶ κατασκηνώσω, but used instead, in Exod 25:8, καὶ ὀφθήσομαι ἐν ὑμῖν, and in Exod 29:45, καὶ ἐπικληθήσομαι ἐν τοῖς υἱοῖς Ισραηλ. (In the other four texts, it gives καὶ κατασκηνώσω, 3 Kgdms 6:13; Ezek 43:9; Zech 2:4, and κατασκηνοῦσιν, Zech 2:5). The reason for its choice of ὀφθήσομαι in Exod 25:8 is presumably to gloss the literal sense by hinting at the future visible presence of the Cloud over the Tent (Exod 33:7–11; 40:34–38; cf. Barrett), and the parallel gloss in 29:45 is intended to stress what the people’s liturgical response to that visible divine presence will be.
All the Targums of these six texts mention the Shekinah: e.g. Neofiti at Exod 25:8 (McNamara), ‘I will make the Glory of my Shekinah dwell among them’, or 1 Kgs 6:13 ‘I will make dwell my Shekinah among the sons of Israel’ (Harrington-Saldarini). The two (post-biblical) terms Shekinah (Hebrew) and Shekintah (Aramaic) both designate the ‘Divine Presence’ among the people of God. One translation including this idea would be, ‘And the Word became flesh, and made his dwelling among us’ (NAB: or, less literally, ‘came to dwell among us’, Knox, NEB).2
It's an allusion to the corresponding texts in Exodus. It may also be very clever wordplay and/or it could be a loanword.