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I'm a complete noob when it comes to guitars, so I have a question. Will any kind of pickup work on a Fender Stratocaster, ignoring the fact that it won't fit in the body?

Other than getting the correct type (single coil or humbucker [pretty much what you noted, getting it to fit]), any will work.

Say I cut a larger hole in my Strat to fit a humbucker. The pickup will work?

It "would." But it's never a good idea. You gotta remember about pole placement and routing depth when you do that kinda thing.