I don't know off the top of my head what the soil capacity is, but I have two wooden boxes that I use for zucchini plants(one in each)that are approx. 14"Wx28L"x4"H. They do fine, but I have to keep on top of watering. I've seen posts by other people that use 5gal buckets. I think your container should be fine.
The container I plant my usual 3 bush cukes in is about 4gal and I use sticks to make a tripod to tie the vines to.
Here's a photo of one of my zuke boxes from a previous season and I used the other box for the cukes that year.
Cukes with supports.