Quit your job. Or, comply. That's your choice. Homeless vagrants don't have to worry about getting a flu shot.

Been prompted to take one myself, but there's no compulsion. I feel bad for anyone who is being compelled to do something they don't want to.

There is a home administered, nasal flu vaccine on the market now, FluMist. You might be able to buy one, show HR the receipt and throw it away.

One thing worth knowing: for a religious exemption you generally do not need a letter from clergy or proof of church membership. Your own written statement of sincerely held belief can be enough, and federal guidance protects personal convictions even ones no organized religion shares.