Not if both parents are American Citizens at the time of your birth!
And what if one of them isn’t?
There are a lot of people out there, especially in our increasingly globalized and mobile society, who are American citizens but might work or go to school for short periods in other countries, and have children while they’re there. So all those children are ineligible for President, while children of illegal aliens will be eligible.
I think people are missing the point that in cases like Arnold Schwarzenegger’s, he legally can’t run, as the Constitution makes clear.
But in a case like Cruz’s, the Constitution really leaves the people to be the “Supreme Court.” It’s public reaction that will determine if he’s eligible or not, oddly enough. And that’s the way it should be, because his being born an American citizen but in Canada, and living his first four years there before coming here, is on the line and something the America people, not a judge, should decide the significance of. So what we say about it will decide it.