Yes. I still think that those with student or work visas or green cards should travel internationally. It's just a question of being smart and carrying the right documents. The right documents will vary depending on the status so I can't give a list but I'd recommend having a brief consultation with an attorney. Many like myself won't charge for a 15-minute consultation, which is all that this should take.
What would a green card holder need to carry other than their passport and greencard?
Maybe just proof of residence in the U.S., such as a lease agreement or deed.
What would happen if they didn’t have that?
I would guess unnecessary delays. Leases or property ownership can be verified with a phone call but probably needs to happen during business hours and if you are prepared everything will have a better chance to go smoothly. Property ownership is public record but a lease might be a bit more trouble to verify if the landlord is privacy conscious.
But I mean what is the conversation like?
“Here are the only documents I am legally required to show you.”
“I need to see a piece of paper with a random address and your name on it.”
“I don’t have that.”
What happens next? They are not able to not let you in.
Unless you're trying to prove a point, having everything they might want will make the process easier. If you want to get into an argument about what you are "required" to show, you may well prevail, but it may cost you six hours.