Dan asks

Hmm, what about other rectangles also with area pi^2/6 ? For example the pi/sqrt(6) x pi/sqrt(6) square.

No, you need one edge of the rectangle to be at least 1.5, if the squares are placed othogonally. It looks even worse if they're rotated 45 degrees. But pi/2 x pi/3 looks like a possibility. Rich