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**. On the other hand the smallest number of squares in a perfect squaring of a square is **21.**   

All perfect squarings of rectangles using between 9 and 17 squares have been found and are listed [here][1].  There appear to be many more rectangles that admit a perfect squaring than squares that admit a perfect squaring. See the quote by David Gale [here][2].

The unofficial [logo][3] 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. 


  [1]: http://www.squaring.net/sq/sr/spsr/spsr.html
  [2]: http://www.squaring.net/sq/ss/ss.html
  [3]: https://uwaterloo.ca/combinatorics-and-optimization/about/combinatorics-and-optimization-square