Is there any reason to change your toothbrush during or after having the flu?

Prolongs illness. You should change your toothbrush about every three months. There is some concern that germs can survive on a toothbrush so you may reinfect yourself and prolong your illness. So it is a good idea to do this if you have the flu.
Get a new brush. Yes, there could still be some bacteria in the bristles and you could be inoculating yourself with these bacteria every time you brush. Just get a new brush after every cold or flu.
Yes, do so. It is also important to change toothbrushes after you've had a cold, the flu, a mouth infection or a sore throat. That's because germs can hide in toothbrush bristles and lead to reinfection.