We are open 24 hours a day 365 days a year. We work 12 hour rotating shifts. When someone takes a vacation/personal day we are randomly put on as coverage. We all have lives and prior engagements and can not always work someone else's shift when it is our scheduled day off due to Birthday, out of town, weddings, etc.... When we tell management we can not do that coverage they get annoyed and tell us it is our responsibility and part of our work duty to cover on our scheduled day off due to someone else who was scheduled took vacation. They look down on people who cant cover.
Is there a law that states that coverage must be done? And could write ups and termination occur if management feels like it due to not doing coverage?

There is no law that says you only have to cover if you want to, if that's what you mean. Yes, you can be written up or fired for refusing.

The employer must cover 168 hours a week. That comes to four folks working 40 hours a week and one covering eight hours or some other combination adding up to 168 hours.

What do you expect to happen when one of them is out for vacation, sick or otherwise not able to work?