If you’re driving around in snow, I can’t imagine you’re driving that fast… so even if the wheel DOES fall off, it won’t be that bad! Seriously, no, your wheel’s not gonna fall off. …probably.
As for the quote of $350… that depends on just what all they’re planning on replacing. If it’s just a couple control-arm bushings, then, yah, it seems steep. If they’re including other work like new tie-rod stuff, etc… then it seems like an ok price.