First off, I want to make clear that I'm not an attorney and what follows is not legal advice :-) If your brother has a court appearance coming up, I would hope he also has an attorney who can clarify this and, possibly, get this appearance rescheduled for him.
Unless this is jury duty or you have been subpoenaed to testify in a criminal case, then it falls under the same umbrella as taking time off for any other personal matter: fixing the plumbing, taking the dog to the vet, etc. and there's no 'job protection' in place.
If it were me, I'd see it as a simple choice: Court over job. Sucks, but the fallout from not appearing in court could be much worse than getting fired.
Two things: You say he is worried he may get fired if he can't move the date. He needs to find out. He should talk to his employer and make it known that he has to go to court, has no options and he wants to know what the repercussions will be.
Secondly, you ask if he can get fired for having court dates. Ohio is an at-will employment state. He can be fired for any reason at all, provided it's not illegal. There are exceptions to this, of course (jury duty, for example) but employees/employers are free to sever ties for any or no reason at all in Ohio.