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 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