I can assure you most H1-B employees would be scared to mess with a good thing. Unless they are well experienced with H1-B or their job offer is quality w/ promises of an immigration lawyer taking care of it.
Yes, several of my friends are "stuck" working for big companies that people on HN wouldn't want to work for until they get their green cards.
They can, but it is significantly complicated and expensive than for a non H1B