Yes, there are non-square rectangles that admit a perfect squaring. The smallest number of squares in a perfect squaring of a rectangle is 9. All perfect squarings of rectangles using between 9 and 17 squares have been found and are listed here.
The unofficial logo of the Department of Combinatorics and Optimization at the University of Waterloo is actually a perfect squaring of the 32 $\times$ 33 rectangle using 9 squares.