My DS had some anxieties when he was about that age. He also worried about his health a lot, excessively. I think that at that age their GT brains pick up a lot of information, but aren't able to process is fully and they tend to get stuck in the worry spots. It may be part of assynchronous development.
A GT counselor suggested we read The Anxiety Cure For Kids. It gave us a context in which to understand his worries and a vocabulary for discussing it within our family. I would not have said that DS had an axiety disorder, but maybe was tending toward a somewhat anxious world view. But once we were able to talk about it and name what was happening in his brain when he started worrying, it really got him back into a more optimistic world view.
