The technically correct term is “illegal alien”.
That is my preferred term for them.
That’s right. An immigrant is by definition legal. A person who is here without the legal right to be here is never an immigrant.
For moe, even that term is too soft. Prefixing an action with "illegal" is used for relatively innocuous things such as "illegal fishing" or "illegal parking". My preference for describing the human flotsam and jetsam flooding America would be "criminal invaders" or perhaps "alien terrorists".
Ah, you pointed it out before I did!