Around four million people who are here unlawfully entered legally; they’re people who could get visas and later violated the terms of them. They are people with slightly more options, because in some cases having overstayed a visa isn’t a bar to becoming a permanent resident from within the country. Some estimated number (anywhere from about two to about 30 million, depending on who you ask; having entered without inspection means no one counted you coming in) are people who came in by sneaking in. Every legal option for them that’s in place now requires them to leave the country first, and usually wait out a ban of ten years.
So, yeah, ditch “illegal” in favor of using words that actually have some meaning.
I think the noun matters. I see nothing wrong with calling an illegal immigrant an illegal immigrant. What I find offensive is the shorthand "illegal" as a noun. It dehumanizes somehow. It defines an entire person as beyond the pale. Literally and figuratively.