I work in a non profit hospital here in Michigan. If my doctor tells me that I can not go to work, either from an illness such as pneumonia, or pink eye, or for some other reason I am unable to do my job, and take care of patiens, are they able to force me to work? Can they fire me for following a doctors orders?

That would depend on whether you, the hospital, and the medical condition all qualified for FMLA.

IF FMLA applies, then you are entitled to up to 12 weeks of medical leave with your job protected.

IF FMLA does not apply, then with very rare exceptions yes, you could be fired for missing work. A doctor's note has no force in law outside of FMLA.