I think the result is false.  Consider a sequence of drawings, one of which I will
represent here:
<PRE>
&&&&&&&&&&&&&&&
&  &  &  &  &
&  &  &  &  &
&& && && && &&
&  &  &  &  &
&  &  &  &  &
&& && && && &&
&& && && && &&

</PRE>
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. (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