I think the result is false. Consider a sequence of drawings, one of which I will represent here:
###############&&&&&&&&&&&&&&& # # # # # # # # # # ## ## ## ## ##& & & & & # # # # # # # # # # ## ## ## ## ##& & & & & ## ## ## ## ##&& && && && && & & & & & & & & & & && && && && && && && && && &&
This is a coloring of a 9 x 15 region which satisfies the conditions and has no 3x3 square with six unit squares of the same color. It(unfortunately, there are some rendering problems as I am not seeing how to control the line spacing.) It should be clear how to extend this for mxn regions in which both m and n are arbitrarily large. Now the idea is to develop a compactness style argument which expresses connectedness of both regions, the lack of a 3x3 subregion with at least 6 squares of one color, and the arbitrary size of the diagram. While I do not have the argument nailed down, I suspect one can use this to show an infinite domain colored in such a way as to preserve all the properties. This (plus other poster's evidence to the contrary) is why I believe the poster's assertion that such a 3x3 square exists that contains at least 6 squares of one color is false.
Gerhard "Ask Me About System Design" Paseman, 2010.09.05