Influenza is a communicable disease, which of course means that there must first be exposure to the virus, which can happen any time of year. In fact, the virus (as well as common cold virus) will literally dry out within days (or weeks depending on the strain) if not in suitable host for infection.
So it's not as if they suddenly reappear during the winter months out of nowhere. The viruses have to be able to reinfect and start the cycle year round. There are assorted theories as to why infection is higher in the winter months, including that people stay together indoors more or that the body reacts differently depending on the temperature which may or may not increase/decrease the chances. But it's never eliminated because of the weather or time of year. And of course there are people traveling around the world, so those from peak flu months could then reinfect when they travel.
My in-laws are nuts about this. The complain that wet hair or even taking a shower in bare feet will lead to illness.