They are probably covered under President Marsdin's Alien Amnesty Act, and are now legal residents, maybe even citizens, depending on how the law is worded.

If I were to go all political I'd say down with borders. But yeah, what Zendar said.

Based on what we know with the exception of Lyra they would be considered refugees, not immigrants. After all J'onn, M'gann, Kara, Kal El, and Mon El came to the US not so much to build a better life for themselves as much as to not die.
This earth's US doesn't appear to have a way of naturalizing immigrants/refugees from other planets. So you can't really blame them for being "illegal" immigrants when they don't have a legal path to citizenship. It's possible that there is a sorta standard protocol that most planets have for off world immigrants, where you figure out the paperwork after you arrive, or something, and Earth as primitive as it is not being capable of interstellar travel doesn't know about these intergalactic norms.

Besides they can't deport them, Not unless NASA's budget, and capabilities have increased tremendously. We don't have the ability to send them back home (with the possible exception of Martians) and even if we could, it would be ridiculously expensive to deport just a fraction of them. And that's only if we assume the other planet is willing to take them back. What If they are refused entry? So the US would have to have diplomatic relations with other planets, which there is no evidence of. (actually along with Fort Rozz that might be the reason why so many criminal Aliens come here, earth probably doesn't have any extradition agreements with other planets either. Earth is probably considered a haven for criminals, runaway slaves or anyone else that might be on the run)

The US can't really force other (earth) countries to take them (after all they aren't from said other countries either). It looks like they're here to stay.