One of the purposes of this site is to combat the stupidity of Political Correctness, a stupidity that in many contemporary liberals, i.e., leftists, is willful and therefore morally censurable. The euphemism 'undocumented worker' is a good example of a PC expression. It does not require great logical acumen to see that 'undocumented worker' and 'illegal alien' are not coextensive expressions. The extension of a term is the class of things to which it applies. In the diagram below, let A be the class of illegal aliens, B the class of undocumented workers, and A^B the intersection of these two classes. All three regions in the diagram are non-empty, which shows that A and B are not coextensive, and so are not the same class. Since A and B are not the same class, 'undocumented worker' and 'illegal alien' do not have the same intension or meaning. Differing in both extension and intension, these expressions are not intersubstitutable.
To see why, note first that there are illegal aliens who are not workers since they are either petty criminals, or members of organized criminal gangs e.g., MS-13, some of whose members are illegal aliens, or terrorists, or too young to work, or unable to work. Note second that there are illegal aliens who have documents all right -- forged documents. Note third that there are undocumented workers who are not aliens: there are American citizens who work but without the legally requisite licenses and permits.
So the correct term is 'illegal alien.' It is descriptive and accurate and there is no reason why it should not be used.
Now will this little logical exercise convince a leftist to use language responsibly and stop obfuscating the issue? Of course not. Leftism in some of its forms is willfully embraced reality denial, and in other of its forms is a cognitive aberration, something like a mental illness, in need of therapy rather than refutation. In a longer post I would finesse the point by discussing the cognitive therapy of Stoic and neo-Stoic schools, which does include some logical refutation of unhealthy views and attitudes, but my rough-and-ready point stands: one cannot refute the sick. They need treatment and quarantine and those who go near them should employ appropriate prophylactics.
So why did I bother writing the above? Because there are people who have not yet succumbed to the PC malady and might benefit from a bit of logical prophylaxis. One can hope.
Hope for the best. But prepare for the worst.