There is a simple proof that a game of [Hex][1] must have a winner, which implies the result you want.See here: [Brouwer's Fixed Point Theorem and the Jordan Curve Theorem][3], Lemma 5.5.
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][2] (by David Gale. The American Mathematical Monthly, Vol. 86, No. 10. (Dec., 1979), pp. 818-827). 


Edit: Actually the reference shows that the Game of Hex always has a winner => Brouwer Fixed Point theorem => a pair of curves in the square joining opposite corners must intersect. So it does use Brouwer's fixed point theorem, but gives an elementary proof of it.


  [1]: http://en.wikipedia.org/wiki/Hex_%28board_game%29
  [2]: http://www.math.pitt.edu/~gartside/hex_Browuer.pdf
  [3]: http://www.math.auckland.ac.nz/class750/section5.pdf