You do have to handle the vines carefully because, as you know, the stems of the leaves are hollow and break easily. The vines are tough enough and hard to damage and the squash themselves have such strong stems that you don't have to worry about them at all.
I also grow canteloupes on a trellis and I'd be willing to bet that those small watermelons would work as well. My yellow squash is more of a bush, so they are not on a trellis. Same with the zucchini.
When it is necessary to tie a vine to the trellis, I use a green velcro on a roll that you can cut to the length you need. Made for plants, it is very strong and re-useable too. I use it to tie my tomatoes and their peppers to their stakes.
Thanks!