There is a simple proof that a game of Hex must have a winner, which implies the result you wantwant.See here: Brouwer's Fixed Point Theorem and the Jordan Curve Theorem, Lemma 5.5. Take a look at The Brouwer fixed point theorem and the Jordan Curve theorem follow from this.
This proof is based on the paper The Game of Hex and the Brouwer Fixed-Point Theorem (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). The
Edit: Actually the reference shows that the Game of Hex always has a winner => Brouwer fixed point Fixed Point theorem and => a pair of curves in the Jordan Curve theorem follow from thissquare joining opposite corners must intersect.
AlsoSo it does use Brouwer's fixed point theorem, there is a short but gives an elementary proof here: Brouwer's Fixed Point Theorem and the Jordan Curve Theoremof it.