Funny, author John Byrne actually hated the fact that Superman is technically not an American so when he got the chance to rewrite the character's origin in his influential The Man of Steel story (the uh, comic not the movie) he added this little tidbit

The series includes the embryonic Kal-El rocketing away from the destruction of Krypton and his birth upon landing in Kansas when he emerged from the artificial womb

Retconning it so that he is born on US soil, making him a legitimate American born in America. This is of course, taken from its Wikipedia* page but you can look up the comic itself and confirm it for yourself. What's even crazier is that the reason he hates Superman not being a real American is that he won't be able to run for President. I'm not kidding.

^{https://en.wikipedia.org/wiki/The_Man_of_Steel_(comics)}

EDIT:

Also yes, Superman being eligible to run for president does become a plot point for a story arc in an event.