Actually, if we look at two half-cycle waveforms, one sine and one square, like this:

http://i.imgur.com/oE5NFZ9.png

(Just the left 1/2 of the graph for this example)

The integral of the sine waveform with a peak value of 1, on the interval 0 x < pi, is :

http://www.wolframalpha.com/input/?i=integrate%28sin%28x%29%...

= 2

The integral of the square wave on the same interval is = pi

So the ratio increase in average voltage at the speaker (comparing the sine to the square) is pi/2 = 1.57. The increase in speaker power is (pi/2)^2 = 2.46.