can an employer ask about religious preference post-hire?

The asking is not the illegal part - the using of that information to discrimnate is.  Its hard to tell from your question, but it appears that you may be an employer.  It also sounds like this clinic may be a vendor of sorts.  Assuming I am correct as to both of the forgoing, I would make sure the vendor never discloses to the employer the employees' answer to this question.  If you don't have the information, it would be much easier to defend a religious discrimination claim brought by the employee at a later date.