The more I research this, the more I am starting to think that Trump might have a point. The Constitution Society has a long, exhaustive article on this that seems to show that Cruz is not eligible
under a rigid reading of the Constitution and English common law, Blackstone, Tucker, etc.:
Presidential Eligibility
However, I think one can also make a strong argument that the framers did not intend for the natural born citizen requirement to apply to people like Ted Cruz, who was born to an American mother and who has lived in America, as an American citizen, since he was 4 years old. The stated idea behind the natural born citizen clause was to prevent "foreign influence" from inserting itself into our government and to ensure that the president would feel no loyalty for another country.
It will be very interesting to see how this plays out.