Can an employer terminate and employee for not getting a flu shot?

Yes, you can potentialy be terminated for not getting a flu shot: an employer is entitled to require this to protect not just patients (though note: even if you have no direct patient contact, you are in a hospital setting, so any flu viruses you introduce into the environment/air could very realistically reach patients) but also other staff members. 
However, the employer should have offered you the "reasonable accommodation," because you claimed a religious reason (though see below for some issues with that). A common reasonable accommodation in this context is to give you the choice of getting the flu shot or, if your faith forbids, alternately, wearing a surgical-style mask the entire time you are at work. If the employer did not give you the option of this or a similar accommodation, you may have suffered illegal religious-based employment discrimination, and you may have a claim to pursue with the federal Equal Employment Opportunity (EEOC). If they did give a reasonable option and you refused both the shot and the accommodation, then your termination would be lawful and you would not have a claim.
Note however that if you do this, you employer may defend itself in the basis that your claimed religious reason is a pretext or an excuse, not genuine. If you just "pray by yourself," it may be very hard to prove that you truly do adhere to a faith which would bar the shot and that it's not the case that you simply don't want the shot for other reasons and are using religion as a justification. So while if you were not offered an accommodation and were simply terminated, you can certainly bring a complaint to the EEOC, be aware that there are challenges or issues with your case.
As for the hours: if an hourly employee, you must be paid for all hours worked and if not could file a wage and hour complaint. This would be filed with  different agency: the departmet of labor, which enforces the wage and hour laws.