In the US, it's very clear. If they don't have work authorization from USCIS, they shouldn't be doing work. There's some guidelines out there on the internet [1], but the students should be extremely careful; any form of compensation or expectation of future compensation for their work could put them in serious trouble. The University of Michigan has a more fleshed out guideline page for their international students [2].
There are acceptable grounds for revocation of naturalization [1], but I'm not sure that having previously violated visa terms necessarily qualifies. The question becomes was he lawfully admitted to permanent residency, and was there a concealment of material facts or willful misrepresentation involved in the process. We would need to really see his immigration file and have knowledge of the dates he was in the US, and when he performed work.