Another distinction, of some importance, to which you pay no heed (why?).
Alien, in today's world, refers to something entirely more different that ourselves. While an alien might be someone from a totally different place (alienation and other such terms illustrate the element of uniqueness or overwhelming difference) mere decades ago, we know today that people from other parts of the continent and even world are not really that different than ourselves. So, the term alien serves to alienate in at least two ways:
1. The word 'alien', today, refers to a different species from outer space.
2. The terrestrial outsider is not as different as we once imagined.
Thus, the term alien serves to marginalize people as "the other, not like us". That's why using the term is not helpful. Immigrants is fine and less of a general term; the term alien is subjective in general context, the term immigrant is not subjective and context is inherently defined.
So, undocumented immigrants. Of course, immigration without documentation is illegal - everywhere. We're not forgetting about any crimes; the term undocumented immigrant is the crime itself in the designation.
One can continue to use the grammatical and contextual (respectively) error "illegal aliens", but it looks stupid to continue after having it explained.