Earlier quoted context omitted.
Why are you scare quoting "illegally"? They are here in violation of the law. If you don't believe in controlled immigration, just state that up front rather than trying to hide behind an appeal to emotion via stories of tearing families apart. Your argument seems to be either that long time violators of a law should gain immunity to the law; or that we should grant citizenship to anyone who enters the country. Which…
Because it is the only case in which "illegal" refers to a status rather than an action, as if the very act of existing in the United States without correct documentation was an affront against justice. This is a very un-American view of how the law works, and so the scare quotes are merited.
That's because the illegal action consists of being in a location where you have no right to be. And it's not, in fact, the only case. Compare it to its close relative, trespassing:
An "illegal resident" is someone living in a country of which they have neither citizenship nor visa.
An "illegal occupant" is someone living in a building of which they have neither possession nor tenancy.
It seems rather straightforward to me. Whether such a view is "un-American" or not I legitimately do not know, not being an American myself. Though I can say that that word doesn't have the proudest of histories, to put it mildly.