I don’t know about the civil lawsuit thing. I don’t think you would have a very good chance of being successful. On the employment side the employer can pretty much make up the rules and as long as they aren’t discriminating against a protected group they are covered. For example a lot of companies have a nepotism policy where you can’t sleep with your boss. If you get caught you could be fired if that’s the policy. It’s not illegal to sleep with your boss and there would be no hope of an action in a civil court but the employer has every right to fire you for it. Same goes with something like a policy prohibiting long hair, facial hair or other form of personal appearance. It’s not illegal to have long hair and a beard but if the employer says it’s their policy they can fire you for it unless you can prove it’s for religious purposes or some other form of protected class.
As far as Covid goes, I think most employers have the policy to protect employees not punish them so I can’t imagine there are very many people who would even oppose a similar rule. I’m not getting into why this virus is different than the flu. Go back and read the 468 pages from before the mom banned that discussion