Yes, I think hospital employees should be forced to continue working when collective bargaining fails. They made the choice to work in a profession that provides care for emergencies, illnesses and diseases. They did not choose the health care field based on income. It should be a given that people in health care would find different options besides work stoppages.
Just like police and firefighters can't strike because lives are at stake, the same should be true of healthcare professionals such as nurses and doctors at hospitals. If someone dies due the negligence of the hospital because of a lack of staff, both labor and management will be sued by the victim's family. Hospital employees should keep working without a collective bargaining agreement because the next agreement will include pay for time spent without the legal contract in place.
The employees who care for the most vulnerable and critical patients should still work when collective bargaining fails or should at least make sure they have some skilled trained fill-ins. For babies in the NICU, patients who are critical and those patients who need almost constant monitoring, continuing care is so crucial.
I do not believe hospital employees should be forced to continue working when collective bargaining fails. If these people are so unhappy with their work that they want to strike, go ahead and let them. There are plenty of other unemployed people in this country who will gladly replace them, probably at a lower wage. Things are too competitive for industries to think that this is actually a problem.
Collective bargaining is one of the most powerful tools we have in our democracy to protect the rights of citizens, workers, and laborers. When strike action like collective bargaining fails, the parties involved should in no way be punished for their constitutionally protected right to assemble and protect their rights.