Melons and cucumbers or squash definately cant cross pollinate. Kind of like a pig and a horse cant breed, even though they both have hooves and tails. They are not closely related enough for it to work, and the only way to really do it would be to modify the genetics of the plant, but through genetic modification, you could cross a melon and a mushroom is you wanted (atleast if you are a scientist). However, squash can cross with squash, cucumber with cucumber, and melon with melon.
Did you cucumber seeds come from your crop last year? Or did you buy them from a distributor? Or did you buy plants? And on your half squash, half cucumbers, what is the flesh like? Is it like a squash or like a cucumber?
Its true that is you save your own, you may end up with a hybrid, depending on what all is planted in your area (not just your garden, because pollen travels). Thats why for fruit trees and things, its best to take a cutting, instead of using seeds from the trees apples. I always liked to see when my plants interbred and what would come up the next year. Some of them were pretty fun.