This is because a lot of French households use electrical heating (because it was encouraged by the government to use the huge sovereign electrical capacity), so cold put a huge stress on the grid. This is not related to nuclear power per se.
It's not entirely unrelated: Nuclear plants make bad peakers, and I suspect the older plants probably also aren't too good at load-following. The French also not only get into trouble when demand is high in winter, but also when it's too hot ín the summer because they don't want to boil their rivers...
