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.  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