The amp won't put out any more power, anything negligable at least.  Even if the coil does act like a resistor for the mS/uS the driver attemps to be at the peak of the sqaure wave (there is ringing, dampening, back emf and all other kinds of things that probably won't let the driver act like a true resistor for that brief period of time) it still would be a resistor with a value a little lower than the overall inpedance of the driver.

I am not certain, but I am also pretty sure that a driver won't behave linearly enough above say 20-30Hz (guessing here) to really produce a square wave, let alone behave linearly enough to produce all those harmonics perfectly.  Drivers still have dampening, the combination of the motor and suspention still shape the way they could attempt to produce a square wave.  

The bottom line, I think, is, square or sine, the driver still moves just as far from +/-, sweeps the same amount of air and cools the coice coil very similary.  And coils themselves (no added forced cooling) could care less what kind of signal is applied....