The emergency rooms and other medical specialties use plenty of immigrants, who go through mountains of paperwork to come in.I fail to see how an illegal could possible get a licese to practice medicine in a major hospital.
I would imagine that the situation is similar with nurses, although the US is facing a crucial shortage of nurses in many places. Hospital nursing is bone-breaking work, with long hours, huge responsibilities, and in many cases, low pay.
I would hope that the paperwork requirements for nurses is equivalent (more or less) to that of doctors–but I am not sure. Certainly in the better and more responsible hospitals it is.