While I was on vacation last week I was browsing through Shrubs and Vines of Vermont by Kate Carter. I had bought it back in the spring to go with her previous book Wildflowers of Vermont, which I heartily enjoyed as a reference. In the section on vines I came across the answer to my mystery vine. The leaves and flowers look exactly like a Wild Cucumber, also called a Balsam Apple (Latin name Echinocystis lobata) and it is on page 145. What surprised me are the fruit, described as a spiny green bladder with dry seeds when ripe. Now the plant I was nurturing is long gone, but I've noticed the distinctive blossoms of the plant winding its way over other plants in the hedge next to my garden. I'll be keeping an eye out to see these distinctive gourds. In some ways this identification is encouraging because it is a gourd which means that we weren't way off in thinking it was a squash of some sort. It's seed leaves and other clues were there because it's in the same family. Also there is a note that says it is often cultivated.
Here is a photo of the female flower with the small beginning fruit. The flowers at the top of the entry are the male flowers. You can see what a mature fruit would look like in a collection of photos of Wild Cucumber and its seeds available at Cal Photos.