[kabar]

Children born in the US of illegal alien parents don't have to go thru the naturalization process; children born to green card holders do not have to go thru the naturalization process; children born to tourists do not have to go thru the naturalization process. They are all American citizens thru birthright citizenship (jus solis.)

Are they all considered to be eligible to be President of the United States under the Constitution, i.e., natural born citizens?

[Yosemitest]

And WHAT DOES THAT have to do with this debate ?
ABSOLUTELY NOTHING !

So there is the law for the time Ted Cruz was born,
AND HOW Ted Cruz's PARENTS fulfilled ALL those requirements of the law that time, for Ted Cruz to be a "Natural Born Citizen".
Ted Cruz did NOT NEED a Court and a Judge to "Nationalize" him.

“Senator Cruz became a U.S. citizen at birth, and he never had to go through a naturalization process after birth to become a U.S. citizen,” said spokeswoman Catherine Frazier.
... The U.S. Constitution allows only a “natural born” American citizen to serve as president.
Most legal scholars who have studied the question agree that includes an American born overseas to an American parent, such as Cruz.