The Constitution requires that a President be born in the USA
No, it does not. It requires that the President be a "natural-born citizen". Citizenship law says that Obama is a US citizen because his mother is one.
However, nowhere in citizenship law is "natural born" defined.
It is possible that a case could be brought that made its way to the Supremes and forced them to define "natural born". Until then there is no definition in law other than its mention in the constitution.
Therefore there is no particular reason to suspect that Obama or Cruz are not natural born citizens, aside from the ludicrous conspiracy fantasy to which Faith and the other nutjobs subscribe.