The 64K$ question is, can the employer fire you for refusing to sign a release of liability?
I would like to think the answer is “no”, but we currently have a Department of Injustice and political prisoners in the US, so, who knows?
IMO: if the employer says they will fire you for refusing to sign a release of liability, well, that smacks of coercion to me, and coercion is illegal.
“The 64K$ question is, can the employer fire you for refusing to sign a release of liability?”
Much of this would depend on your state. If you live in an at-will employment state the answer is yes.
In other states it would depend on the rules for when someone can be fired.
The only exception would be if you could come up with a religious exemption, but that is unlikely.