The born-again experience means a Jew is no longer a son of Israel and a Gentile is no longer a Gentile. Both have become sons of God—a new man.
Throughout Scripture the term "son of God" is reserved and applied only to those who have been or are created by God's direct actions.
Adam was a son of God (Luke 3:38) (as was Eve by definition), celestial beings are referred to as sons of God (Job 1:6), Yeshua (Jesus) was the conceived by Holy Spirit—a direct act of God (Luke 1:35), and we who are born of the Spirit as a result of Yeshua's sacrifice (Galatians 3:26).
It seems to come down to the point that we are either a follower of Yeshua—the new man—or we are are a Jew, a Gentile, or some other identifier.
"There is neither Jew nor Greek, there is neither slave nor free man, there is neither male nor female; for you are all one in Christ Jesus." (Galatians 3:28 NASB unless noted)
"For in Christ Jesus neither circumcision nor uncircumcision means anything" (Galatians 5:6)
"you laid aside the old self with its evil practices, and have put on the new self who is being renewed to a true knowledge according to the image of the One who created him—a renewal in which there is no distinction between Greek and Jew, circumcised and uncircumcised, barbarian, Scythian, slave and freeman, but Christ is all, and in all." (Colossians 3:9-11)