Asylee is a specific term, it isn’t the same as an applicant:
Asylum applicants can’t legally work except under certain conditions.
I think it is safe to assume this about asylees with a legal right to work and that have certain legal protections from discrimination.
Reread the OP, he referred to them as illegals.
Of course not.
That is true.
Is there any indication of that with SpaceX? SpaceX has hired H-1B applicants for example (Musk himself was a H-1B recipient).
This site gives a more complete account of why SpaceX is being sued and notes this:
Under the INA, employers cannot discriminate against asylees and refugees in the hiring process, unless there is a law, regulation, executive order, or government contract that requires the employer to do so. The IER is charged with enforcing the anti-discriminatory provisions of the INA. The INA generally prohibits discrimination based on citizenship status and national origin when hiring, firing or recruiting.