But is diarrhea a required symptom of stomach flu?

Not at first: When you first get "a stomach flu bug", your gut may protest it by trying to shut down altogether. In this stage you feel the nausea and vomiting, because stuff just does not want to go through. It's usually later that the body tries to "wash out the virus" by dumping more fluids in the gut to try and wash out the virus - and then boom, diarrhea. So early on you may not poop at all.