“Starting” a nuclear (or any war) is not necessarily the intention of anyone…not any person we’d generally agree is sane in the general use of the word. But actions -> reactions; and as we’ve seen, war is too often the outcome or unintended consequence of actions that might have been seen at the time by the acting party to be “safe” and/or without consequence.
When one nation sees another as attempting hegemony and/or otherwise threatening their national security, war is not too far away.
Clinton’s belligerence towards Russia was well-documented, whether with respect to Ukraine / Crimea or to Syria, where she pushed for a no-fly zone that many military observers felt would likely lead to a direct conflict / encounter between Russian and American air forces.
Those were the reasons why some people feared she could “start” a nuclear war, as in help create the escalating tensions that would increase the possibility.
While there are always those who oversimplify and use hyperbole in their internet posts (the nature of a sound-bite culture, I’m afraid), the increase in US-Russian tensions associated with Clinton’s policies (and her inner circle of advisers) was a reasonable fear - in my opinion and in the estimate of many more expert in these topics than I. It wasn’t “fear mongering” at all…it was the articulation of very valid concerns based on the evidence.
In contrast, Trump claimed and previously had indicated a desire for a better relationship w/ Russia. Given his nature, I personally was under no illusion that his actions would match his words. However, it’s understandable to me that some people would give him the benefit of the doubt on that one point when the other candidate was clearly and openly hawkish.