there is a very good apoligetics page for this subject on the main Catholic Answers site.
as for my view, I would question them on the meaning of the passages where the phrase occurs, on of which is the beginning of the third chapter of John
Joh 3:1 And there was a man of the Pharisees, named Nicodemus, a ruler of the Jews.
Joh 3:2 This man came to Jesus by night and said to him: Rabbi, we know that thou art come a teacher from God; for no man can do these signs which thou dost, unless God be with him.
Joh 3:3 Jesus answered and said to him: Amen, amen, I say to thee, unless a man be born again, he cannot see the kingdom of God.
Joh 3:4 Nicodemus saith to him: How can a man be born when he is old? Can he enter a second time into his mother’s womb and be born again?
Joh 3:5 Jesus answered: Amen, amen, I say to thee, unless a man be born again of water and the Holy Ghost, he cannot enter into the kingdom of God.
Joh 3:6 That which is born of the flesh is flesh: and that which is born of the Spirit is spirit.
Joh 3:7 Wonder not that I said to thee: You must be born again.
Joh 3:8 The Spirit breatheth where he will and thou hearest his voice: but thou knowest not whence he cometh and whither he goeth. So is every one that is born of the Spirit.
this sounds more like a discussion on baptism, though Nicodemeus would not have realized it.
the other major occurence is
1Pe 1:22 Purifying your souls in the obedience of charity, with a brotherly love, from a sincere heart love one another earnestly:
1Pe 1:23 Being born again, not of corruptible seed, but incorruptible, by the word of God who liveth and remaineth for ever.
which seems to talk about love, charity, and obedience. strangely it also says that we must purify ourselves through obediently doing the works of love. I would suggest here that “word of God” is probably Christ and not the bible which the english structure here supports (though I do not know enough greek to say if the origional construction is indicitive of the same)
personally i think that the warnings of Peter apply
2Pe 2:18 For, speaking proud words of vanity, they allure by the desires of fleshly riotousness those who for a little while escape, such as converse in error:
2Pe 2:19 Promising them liberty, whereas they themselves are the slaves of corruption. For by whom a man is overcome, of the same also he is the slave.
and
2Pe 3:15 And account the longsuffering of our Lord, salvation: as also our most dear brother Paul, according to the wisdom given him, hath written to you:
2Pe 3:16 As also in all his epistles, speaking in them of these things; in which are certain things hard to be understood, which the unlearned and unstable wrest, as they do also the other scriptures, to their own destruction.
it sometimes seems that some of this who dwell on “born again” (as well as “assurance of salvation” which is another conversation altogether) use these concepts as excuses not to delve deeper into what scripture asks of us.
by the way, notice it is Peter who warns us against twisting the scriptures to our own destruction, perhaps it is inevitable that they should ignore him as they ignore his successors