Some illegals have college degrees.
I can’t speak for other companies, but having worked in several companies and STILL working for one, I can attest that the FIRST THING the HR department checks is your PERMANENT RESIDENCY (i.e. Green Card), CITIZENSHIP or WORKING VISA.
I do work with Indians, Chinese and Filipino software developers ( all with college degrees ), but ALL of them either have H1B visas or are on intra-company work permit. Which means they are NOT ILLEGAL.