do I have the right to talk to security myself, if my manager refuses? Sure. No laws prevent you from doing this. But I (a) doubt security is going to do anything contrary to management's decision and (b) you may find your bosses not very pleased with you that you went around then and tried to take matters into your own hands.
Can they fire me for my concerns for my safety? Yes, if you are insubordinate and disregard how management has decided to handle the situation.
My husband wants to file a restraining order (not sure if that's overboard or not), but how would that work with me working there and their child being there, with them needing to visit? Your husband is not a party to this and he can't file for a restraining order on your behalf. And yes, I think that is going way overboard. Even IF a judge agreed to a RO (which does not appear at all likely), how, exactly is that going to work? Either the family cannot come and see their dying child when you're at work or you can't work when they're there.
What kind of legal rights do I have to feel safe at work? Not too much, actually. Apparently this family has a history of being difficult and perhaps making theats before - which has never been followed through on. I don't know what exactly they said since you haven't shared it but I don't see any reason to believe that you are actually in any danger. You filled in for the primary care nurse for an hour. I can't imagine you'd been the family's focal point for their grief when the child dies.
A person, who is nice to you, but rude to a waiter, is not a nice person. (This is very important. Pay attention. It never fails.)