As said by @robert bristow-johnson, a squared window is another window. If you take a quite standard type, symmetric, unimodal, positive (with the related question Conditions for symmetric and unimodal windows in both time and frequency domains) and peaking at $1$, like a Hamming window, you can see the following behavior:
The squared one is below the original, due to the contractive effect of $t \to t^2$ in $[0,1]$ (in other words, $t^2\le t$), while keeping the same maximum location. So the window is better localized, and gets improved apodizing at the ends, which can be useful to reduce "blocking" artifacts at the edges of overlapped frames: if the original window ends at $0.1$, its squared version ends at $0.01$, a ten-fold reduction.
The other effect is that the new window is close to energy-normalized. Even if your "window" is zero-mean (like a wavelet), the energy of its square is never zero (unless the window is zero), so you can use it as a weighted average, of the shape (similar to equation 4):
$$\frac{\sum a^2(k).x(k)}{\sum a^2(k)}\,.$$