Can a laid off employee work for free in order to stay busy and help the company? The employee insist that they work for free.

This sounds like a potentially messy can of worms. I would advise against it - maybe this laid off employee can find a way to stay busy through volunteer work just not at their old company.

I could see the employee getting angry if new employees are hired after they've been laid off. If this situation was in my office I'd say no and wish them well on their way.