Given a countable coloring of the plane, is it always possible to find a monochromatic set of points $\left\{ \left(x,y\right),\left(x+w,y\right),\left(x,y+h\right),\left(x+w,y+h\right)\right\} $ (the corners of a rectangle)?
This is equivalent to CH.
Quoting "Problems and Theorems in Classical Set Theory" by Komjath and Totik, chapter 16, Continuum hypothesis:
CH holds if and only if the plane can be decomposed into countably many parts none containing 4 different points a,b,c,d such that dist(a,b)=dist(c,d)
This is a stronger requirement than your problem, so assuming CH the answer is no. Their solution, assuming CH is false, proves that there's a monochromatic rectangle.
Previous version, with added explanation about Hamel basis:
Using
CH holds if and only if R can be colored by countably many colors such that the equation x+y=u+v has no solution with different x,y,u,v of the same color.
This gives a negative answer assuming CH. Explanation: consider R as a vector space over Q. Let A be some basis. Take any bijection A > A + A, where + is disjoint sum. It induces a linear isomorphism f: R > R * R. (You can think that there's a linear isomorphism between reals and complexes if that helps.) Then, if you were given a monochromatic rectangle a=(x1, y1), b=(x1+x2, y1), c=(x1, y1+y2), d=(x1+x2, y1+y2), certainly a+d=b+c. Using that isomorphism, f(a)+f(d)=f(b)+f(c) gives a monochromatic solution of quoted equation.

$\begingroup$ Could you explain what you mean by "use Hamel basis to represent R as R+R"? $\endgroup$ – Ian H Oct 11 '09 at 23:46
Take a look at this site: http://blog.computationalcomplexity.org/2009/11/17x17challengeworth28900thisisnot.html