I've been thinking about a very similar issue: I was considering giving a talk about the Brouwer fixed point theorem to some math majors (but not necessarily ones very familiar with point-set topology).

There is an elementary proof which uses Sperner's lemma; see Michael Henle's book A Combinatorial Introduction to Topology for details. You can outline a proof of Sperner's lemma pretty quickly (induction on dimension, and dimension 1 is easy), and from that, you can wave your hands (or be more precise, depending on how much your audience knows about compactness) to get Brouwer's theorem. Since Sperner's lemma holds in higher dimensions, you can prove the fixed point theorem in higher dimensions, also.