It depends on how you define "correct," I suppose. If you're using it as a term of art, or you're a right wing think tank, then the correct term is "illegal alien." [0] It does not actually appear in Federal law, however. [1]
The IRS uses the term "undocumented alien," which is kind of a weird, mixed construction, referring to people as "aliens" (which I always find weird, but, okay), but not "illegal." [2]
Other government agencies, and, yes, immigration advocates, use the term "undocumented immigrant," which has the virtue of both being accurate, and not referring to individuals as "illegal," when the thing that's actually illegal is the fact that they are in the country without authorization (the "undocumented" part).
In summary, "illegal alien" as a term of art: fine in my book, just weird. "Undocumented alien": sure, if you're the IRS. But, otherwise, "undocumented immigrant" is the most technically accurate, because it's not the person who is illegal, as the word "illegal" modifying "immigrant" in the phrase would indicate, but their presence in the country that is illegal.
> Other government agencies, and, yes, immigration advocates, use the term "undocumented immigrant," which has the virtue of both being accurate
“undocumented alien” is arguably more accurate; alien is just a (somewhat dated outside of law) term for a foreigner, “immigrant” has further meaning of seeking to make the country their permanent home, which is not always the case for foreigners present without current documentation of legal presence.
