Cross pollination is likely to happen, but unlikely to affect the fruits, although it would certainly affect the genetics of the seeds (which you don't eat.)
However, and this is something I've only even heard of today, squash cross pollination can result in very bitter tasting fruits, which can cause food poisoning and possibly have even longer term effects too.
Wild squashes contain a lot of the chemical causing the problem (cucurbitacin), which is why all the squashes we grow here have been bred to contain minimal levels of it.
Look up 'Toxic squash syndrome' for further information, but don't worry about it unduly - If it was a common problem in this country I think we'd have heard about it by now - advice seems to be to spit out any squash (or dish containing squash, like soup,) which tastes unusually bitter.
