I think that logic and, more broadly, philosophy would be absolutely excellent additions to the high school curriculum, but you really do not need any formal knowledge of mathematical logic to do Euclidean geometry. It's a great introduction to the basic idea of math, starting with a set of facts and deducing a conclusion. People have intuition about geometry in the plane, and introducing more formalism would just obscure everything.
The temptation once you know a lot of math is that math should be a sequential subject where you build up from axioms. If you actually try to teach using this method, by which I mean heaping formalism on students before they need it, you'll find out that it just doesn't stick. That isn't even really how mathematicians do math. Usually the intuition leads to the formalism. People were working with the natural numbers for thousands of years before we decided to nail it down with ZFC.