We really need to face up to the fact that there is currently no method for really proving where someone was born, and some people — particularly the offspring of illegal immigrants — may really not know themselves. If we want to have a way of proving this, we have to accept that it will require the implementation of a huge biometrics-based federal database, along with a legal requirement that births be entered into the database within 24 hours or even less, and that even with that, it will only be functional for births that occurred after the datatbase was fully operational, and there will still be a few cases where a woman who just gave birth runs across the border, shows up at an emergency room saying she just gave hirth in the US (though with a 24 hour requirement, there would be ample opportunity for investigation of suspicious claims.
An excellent reason not to give citizenship to children whose parents are not legally in the country, regardless of where they are born.