Well, technically he is an illegal alien.
Refugee.
None of those opposed to the current immigration problem has any problem with accepting refugees.
But you *were* funny.
Somehow I don't think the new fence they're trying to get built is gonna help a whole lot.
I could see an uproar if this happened in a Captain America movie, but Superman? Superman saves the world over and over, not just America.
"Well, technically he is an illegal alien."
Nope, in the current storyline in the comics his ship was an artificial womb and technically he was born in the USA when it landed and opened in front of the Kents.
As for the American Way stuff, the context is that Perry White is giving orders to different departments and among that he tells them to find out if he still stands for 'Truth, Justice, and all that stuff.' It doesn't come across as a deliberate evastion on Perry's part, but it is on the writers part.
As for the movie itself:
Since Superman is an illegal alien, he should return to his country or planet or origin and come back when he can enter legally.
I was going to post the thought but dithered.
He came here as a very small child and was taken in by a loving American family that lived in Smallville. Later as an adult he as Calrk Kent moved to Metropolis.
It will take a better historian than me to assert absolutely that he was not in fact legally adopted while very very young and there by made an American citizen.