Does this order make them legally responsible for any after-effects that the flu virus may incur.
Probably not.
Here's the problem: Current, provisional CDC guidance about H1N1 "swine" flu makes hospitals and doctor's offices responsible (and therefore liable) for persons who contract swine flu (or who claim they do), but not regular flu, while on the premises.
Under those circumstances, mandatory vaccine for employees is being pushed, hard, by hospital lawyers and insurance companies.
Not without reason.