I also use the EZBox as well as Gary Fong Domes. I would not mount the flash attached to the ezbox on the camera's hot shoe because it would be too cumbersome.
What flash do you have? I purchased and sent back GF collapsible dome for the SB-900 because I had a difficult time getting it on my flash. In fact, I never was able to get it on my flash at all. I instead purchased GF universal dome.
I have the 24 X 24 size. If I mounted it on the hotshoe, the lower front edge of the softbox would be in front of the lens and you wouldn't be able to take any pictures. This is definitely off camera light stand gear.
John Herrel Nikonian from South Carolina See the light, capture the essence!