Alien --

"from another country" -- Merriam Webster.
 "a foreigner" --

This is the easiest definition so far.  But progressives  don't use it because someone might feel offended.

They don't want it to apply when attached to the word "illegal", as in "illegal alien".  Sorry progressives, that's what we call the border-crossers who don't go through the proper immigration channels.

It's not offensive, unless you think truth is offensive. 
It's not unfairly deprecating, unless you feel true words actually unfairly diminish people.
It doesn't contain any dislike of the individual, unless you consider distant words a "micro-aggression".

Want another definition:

Alien: relating, belonging, or owing allegiance to another country or government - Merriam Webster

Well, the illegals here certainly don't owe allegiance to the USA yet.  Especially when they refuse to learn English.

Progressives who don't want to call something by what it is are what is known as "politically correct".  Political correctness is cowardice.

So when Mrs. Clinton claims she will no longer use the term "illegal alien", she is bowing to cowardice, sucking up to voters, or or actually thinks some brazen lawbreakers are offended by her speech.  Progressives do this to feel better about themselves.  But in the process of calling something by an inappropriate name, the put themselves into an even less-trusted category.