Windowing function squared?
it's just another window.  usually this happens when a window is applied on the time-domain data going into the STFT (which is a DFT), some process happens to the frequency-domain data after the STFT, then there is an inverse DFT bringing it back to the time domain, and the window is applied again to that output time-domain data.  if that process done to the data in the frequency domain happens to be the null process, then what comes out of the iDFT is exactly what went into the DFT.  so then, in that case, the window is applied twice and that is the same as applying the square of the window once.

i have been having a little argument with someone on comp.dsp about this.  there is a term i hadn't seen before ("Princen-Bradley condition") which i think means the same as a "complementary-power" window.  if the square of a complementary-power window is applied to an STFT system above with a null process, the output will be exactly the input when 50% overlap is used.  but if the square of a complementary window (such as the Hann) is used, then, with 50% overlap you do not get perfect reconstruction.  so the square-root of a Hann window would be a good candidate for a complementary-power window.  and you would have to apply that window twice to the time-domain data to get perfect reconstruction.