Paul almost certainly was aware from the other apostles of the term "born again" and what it meant, a la John 3:3, James 1:18, 1 Peter 1:23.
But in all his letters, Paul never once was inspired to use it.
In the Gospels/Acts period, one could be born again, or one could be reckoned as co-dead with Christ and raised to newness of life. Not both.
It is a mistake to look merely for a same word use.
This is an issue, not of the things that differ between the Body and Israel, but of one of the various things both share in common.
Both the born again Israelite (one day), and the member of God's New Creature: The Body of Christ (here and now), are the issue of regeneration.
Both of which share a same origin, working in one's inward man, and so on....
Both of which share The Spirit of God in the spirit of men, in common.
Paul does bring up the Body's connection with Israel's New Testament - in 2 Cor. 3, etc.
The Spirit of God.
(though what Paul was actually addressing there was their [as with Law's] "outward appearance...after the flesh" legalism towards him).
But both agencies share regeneration in common and by the Spirit.
Ephesians 2:18 For through him we both have access by one Spirit unto the Father. 2:19 Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God;
The saints there are the Israel of God.
The household of God there is comprised of both agencies: The Israel of God and the Body of Christ.
Both are fellow citizens with one another.
2:20 And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone; 2:21 In whom all the building fitly framed together groweth unto an holy temple in the Lord: 2:22 In whom ye also are builded together for an habitation of God through the Spirit.
No surprise then that Paul quotes the following New Testament promise operating principle in the following...
2 Corinthians 6:16 And what agreement hath the temple of God with idols? for ye are the temple of the living God; as God hath said, I will dwell in them, and walk in them; and I will be their God, and they shall be my people. 6:17 Wherefore come out from among them, and be ye separate, saith the Lord, and touch not the unclean thing; and I will receive you, 6:18 And will be a Father unto you, and ye shall be my sons and daughters, saith the Lord Almighty.
That is the same operating principle as in Ephesians 2.
Ephesians 2 is the issue of the unity of both aspects of God's Two-Fold Purpose: Prophecy (Israel over the Nations of the Earth) and Mystery (the New Creature over God's Heavenly Host).
Which is the Two-Fold Issue...
Ephesians 1:10 That in the dispensation of the fulness of times he might gather together in one all things in Christ, both which are in heaven, and which are on earth; even in him:
The agency in both is The Spirit of God.
Rom. 14:5; Rom. 5:6-8.