And then you died.

... Not a great strategy.

The birth control one is a bit weak, as they would counter with "that is not for a contagious disease". So a better counter is to ask them to produce the documentation that demonstrates isolation of the virus (this does not exist), a description of the ingredients in the vaccine (this does not exist), a long-term double-blind peer reviewed test of the efficacy and safety of the vaccine (this does not exist), and the FDA approval of this vaccine (this does not exist). They should also point out in your employment contract where they can force you to take a medical treatment (this does not exist), and especially an experimental medical treatment (this does not exist), and explain to you how requiring your disclosure of your medical information does not violate HIPAA laws (it does violate HIPAA laws).

If you can work from home, the compromise could be that from now on you will work from home until the policy changes. If you cannot work from home you need to NOT QUIT, and force them to fire you, so you can collect unemployment and sue them for wrongful termination. If they fire a bunch of you, start a class action lawsuit against them and that will absolutely TERRIFY them. But be aware that a class action can take 10+ years.

With this stuff the best approach seems to me to be to come on HARD and demonstrate you are not a pussy and know your rights and want EVERYTHING in writing (this will send the right message), respond with letters that are written by you plus a lawyer (this will scare them), and try to exit with the biggest settlement and go get a better job at a non-mandate place.

IANAL.