>>> Did she ever give up her U.S. citizenship? I think you have to return and spend a certain amount of time back in the U.S. to retain that right.
I don’t know if she did, or if that is a requirement. I doubt it in both cases. However, citizenship is completely ignored where the father is concerned, making it rather irrelevant.
For birth abroad, in wedlock, to one citizen parent, the sex of the citizen parent (and thus of the non-citizen parent) isn't relevant. The citizen parent can be the father, or can be the mother. There is a different treatment between father and mother, but only if the birth is OUT of wedlock.
If we look at the Cruz case, the citizenship and US residence of Mr. Cruz, the father, isn't relevant; the citizenship and residence of the citizen parent, Mrs. Cruz, is relevant.
For the citizen parent to be able to descend citizenship to his or her children, the citizen parent must have been a resident of the US for certain time periods before the birth of the child.
In Cruz's case, in order for his citizenship to persist, he (the child) is ALSO required to meet US residence requirements (Section 301(b) of the 1952 naturalization Act), specifically a 5 years of continuous US residence (a 6 month absence from the US does not disturb continuity) between the ages of 14 and 23. This aspect of the law has since been repealed, but Congress may add it back in, or change the parental residence requirements (it has changed both of these over the years) associated with descending the citizenship in the first place.
Cruz hasn't proved to the public that he is a citizen in the first place, but I think he is.