Interesting question. For semiconductors in general you typically see an increase in the bandgap under increased hydrostatic pressure. So you will see people making experimental measurements to measure the deformation potential. I am not really a theory guy, but maybe one way to think about this is that you have a periodic potential for your crystal and that the fourier components of the potential will end up being related to the band gap. There is a nice set of notes that outline the basics of band theory. In your case perhaps you can make an analogy that way. 

 http://www.phys.utk.edu/courses/Fall%202009/physics671/chapter4b.pdf

That walks through the free electron model, the applies perturbation theory etc and shows this for a simple model.

One very hand wavy way to think about this might be that if you considered some periodic arrangement of atoms, and now that you compressed them now the wavelength of the electron wave  that would be able to travel in the conduction band would shorter and  have higher frequency.  In the case of quantum dots this would be the equivalent of making the box smaller. 

In general, I think that you could also say that as you bring the atoms closer together and you have more interaction you have increased splitting of the energy levels. Depending on the configuration of the energy levels and the resulting band formation you probably mostly see an increase in the band gap  around the gamma point  except in special cases.