Quick answer: False
Long answer. There are several scriptures which address the issue of being “born again in the flesh.” King David had a baby son who died after being ill. David tells us this: 2 Samuel 12:22 “While the child was alive, I fasted and wept; for I said, ‘Who can tell whether the LORD will be gracious to me, that the child may live?’ 23 But now he is dead; why should I fast? Can I bring him back again? I shall go to him, but he shall not return to me.”
David said he will die and go to his son, but the son will not return. If you have lost a baby, how nice to know s/he is in heaven. This would include babies that have been aborted.
Another passage is often brought up to support notions of reincarnation. John the Baptist was selected before he was born to work as Elijah did. Luke 1:16 He (John) will turn many of the children of Israel to the Lord their God. 17 He will also go before Him in the spirit and power of Elijah, ‘to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just, to make ready a people prepared for the Lord.” This is NOT the old Elijah in a new body. In fact Elijah shows up later in front of Jesus and the apostles, looking like himself. Matt 17:4 Peter said to Jesus ….. if You wish, let us make here three tabernacles: one for You, one for Moses, and one for Elijah.” Clearly they knew exactly who Elijah was, not a combo of John and Elijah and any 3rd person either.
Finally, the concept of reincarnation makes the “person” into a bi-polar or worse, multi-personality. If you lived as Hazel, Mary, and Joanne, what are you going to be – 3 personalities? Really?!
The bottom line is that Jesus died for your sins, here and now, not in a future life. He can’t do both. And when you die, once for all, you face the judgment. Hebrews 9:27 It is appointed for men to die once, but after this the judgment
You will need Jesus’ sacrifice then, not at some future time. We know this teaching is false because it reduces the person and work of Christ to nothing.