Faulty Premise #1: John the Baptist was a reincarnation of Elijah
When Jesus asked Peter who people said he was, he answered that some people thought that Jesus was Elijah come back. Peter knew better and said Jesus was the Christ. In any event, John the Baptist himself directly denied the claim (see John 1:19-21).
What Jesus more likely was saying in Matthew 11 is that John the Baptist was a prophet in the tradition of Elijah. Elijah was the greatest of the prophets (even if his disciple Elisha inherited a double portion of his spirit). All of the prophets that followed came in the name and spirit of Elijah, in the same way that some of Paul's letters apparently were created after his death. They were in the spirit and tradition even if not the actual man.
Note also Elijah never died - he was taken up into heaven, and people thought he would return. As such, no Elijah was not being thought to have reincarnated- only returned. Elijah's return was long a part of Jewish custom. Even today, during a Passover sedar, a chair is left empty for Elijah. But again, since he did not die, Jews look for his return, not his reincarnation.
Faulty Premise #2: The Jewish world view has any concept of reincarnation at all
The most basic rule of hermeneutics suggests that a text cannot mean what it could not mean to the original audience. If the Jews had no concept of reincarnation, then any interpretation that says it does has a very, very high bar to clear.
In his book The Gift of the Jews, Thomas Cahill advances the thesis that the primary distinctive philosophy of the Jews was precisely that history was not circular. Their linear idea of a human life was a departure from others and was an explicit rejection of reincarnation. As such, a text would need to be exceedingly explicit to a culture that rejected circular life patterns, and the obscure interpretation above (especially in light of the more obvious intepretation) simply doesn't hold water.
Furthermore, the book of Hebrews - written to the people into which both Jesus and John the Baptist were born - explicitly reject reincarnation. As the writer points out in 9:27,
It is appointed unto a man once to die and then the judgement.
Here we do have an explicit rejection of reincarnation by the followers of Jesus. There is one death. There is one judgement. Hence, in any Christian worldview, you don't come back.