Being a waitress sucks. You have to deal with people all day, and some of them are jerks who will complain about everything.
Management can certainly ask this employee to remove the bracelet. Free speech doesn’t enter into it.
But here's the thing: no matter where one works, there is no such thing as free speech on company time. In the employers eyes, you're there to do the job they hired you for, not express one's own opinions.
Yes, the customers were likely liberal arseholes, chances are the employer was looking for a reason to fire this waitress anyway. If she were any good, likely he'd have simply placated the "customers" until they left and asked her to simply remove it. There has to be more to this story than meets the eyes here.