Why is the Hamming and Hanning window the most widely used for speech spectograms?
any window (almost any) is better than a boxcar window, unless your spectrum is flat.  speech signals are not flat. 

I suspect that Hamming and Hann are commonly used because you can avoid scalloping loss for 50% overlap. One gets a sufficiently favorable tradeoff between time resolution and frequency domain dynamic (side lobe level) range. There is also the fact that these windows are cited in papers that go back to the 70s.  

You can also apply the widows with a 3 point convolution (3 coefficients instead of N) in the frequency domain, which might have had some advantages in hardware in the past when memory was a constraint.

Classical fixed windows are chosen according to characteristics like sidelobe level. They aren’t like adaptive windows that use the data itself.  It’s relatively pedestrian processing. 

in summary, boxcar windows are for flat spectra.