The thing as per the 1952 statute he is eligible:
http://uscode.house.gov/view.xhtml?path=/prelim@title8/chapter12/subchapter3&edition=prelim
:§1401. Nationals and citizens of United States at birth
(d) a person born outside of the United States and its outlying possessions of parents one of whom is a citizen of the United States who has been physically present in the United States or one of its outlying possessions for a continuous period of one year prior to the birth of such person, and the other of whom is a national, but not a citizen of the United States;
(g) a person born outside the geographical limits of the United States and its outlying possessions of parents one of whom is an alien, and the other a citizen of the United States who, prior to the birth of such person, was physically present in the United States or its outlying possessions for a period or periods totaling not less than five years, at least two of which were after attaining the age of fourteen years:”
"By the Thirteenth Amendment of the Constitution, slavery was prohibited. The main object of the opening sentence of the Fourteenth Amendment was to settle the question, upon which there had been a difference of opinion throughout the country and in this court, as to the citizenship of free negroes, Scott v. Sandford, 19 How. 393, and to put it beyond doubt that all persons, white or black, and whether formerly slaves or not, born or naturalized in the United States, and owing no allegiance to any alien power, should be citizens of the United States, and of the State in which they reside. Slaughterhouse Cases, 16 Wall. 36, 83 U. S. 73; Strauder v. West Virginia, 100 U. S. 303, 100 U. S. 306."
"This section contemplates two sources of citizenship, and two sources only: birth and naturalization. The persons declared to be citizens are 'all persons born or naturalized in the United States, and subject to the jurisdiction thereof.' The evident meaning of these last words is not merely subject in some respect or degree to the jurisdiction of the United States, but completely subject to their political jurisdiction, and owing them direct and immediate allegiance. And the words relate to the time of birth in the one case, as they do
Page 169 U. S. 725
to the time of naturalization in the other. Persons not this subject to the jurisdiction of the United States at the time of birth cannot become so afterwards, except by being naturalized, either individually, as by proceedings under the naturalization acts, or collectively, as by the force of a treaty by which foreign territory is acquired."
To be "completely subject" to the political jurisdiction of the United States is to be in no respect or degree subject to the political jurisdiction of any other government.
The fact that Cruz's citizenship depends on a statute makes him naturalized. The touchstone for "naturalized" isn't a naturalization ceremony. All the case law, and it is voluminous, uses the framework that naturalization is making an alien into a citizen, by any means whatsoever. But for the statute, Cruz would be an alien at birth.
“The thing as per the 1952 statute he is eligible:”
For citizenship, certainly, but a statute cannot over ride the Constitution in no form or fashion, does not change the Constitutional requirement to be natural born or what natural born signified in 1789.