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I actually think that Hilbert's Third problem is one of the explainable for school guys. It's even more cool that it exists in such a famous list close to the problems that are so tempting and not yet solved. The question is: can one cut the cube in some polyhedral pieces, reglue them and get a regular tetrahedron? The answer is no and the theorem was proved by Dehn using so-called Dehn invariant. It uses some algebra and number theory but can be understood by high-school level guys. The time you need to explain this is 3-4 hours, so maybe it could be a little and nice course. See, for example, Lectures on Discrete and Polyhedral Geometry by I. Pak