This reference is certainly pertinent, being the second Google hit for "covering a square with squares" (after your question). Just reading it now...
http://www.uccs.edu/~faculty/asoifer/docs/untitled.pdf
UPDATE: so far as I can tell from looking at this article, the author regards your question as an unsolved problem. There is a further article by him and Karabash, apparently in (his) journal Geombinatorics, vol. 18, which I cannot access online and which has not been reviewed on MathSciNet.
