The auto insurance will cover, but the health insurance would also cover anything the auto didn't, in particular if your treatment exceeds your auto insurance limits. If you don't have your own insurance, by law, you have to go through your boyfriend's policy so you aren't doing anything wrong.

Now later on you *might* sue your boyfriend's policy for a variety of reasons. If the other vehicle was at fault you would sue them, but if their coverage limits are lower than what your suit is asking for you could also sue your boyfriend's policy for the rest of it. Alternatively if your boyfriend was at fault you wouldn't sue the other vehicle, in that case you would just sue your boyfriend's policy. In the first case it shouldn't cause a premium increase, in the second case it would, but if he's at fault his insurance is going to go up anyway.