You don't say what sort of square footage you're talking about, but with the many varieties you've mentioned, unless you're planning on picking a couple and only growing one plant a piece, you may be pushing trying to do this as container gardening. Certainly I wouldn't put more than one plant to a cage. And not sure about the kabocha, but both butternut and spaghetti can be rather big and HEAVY. Delicata is wonderful for being small (and personally, I think the sweetest of all squash) - it might work best for you.
However - my suggestion is to get REALLY strong hardware cloth with 4 -6" openings. Make sure it's anchored deeply into your container. Plant three seeds around the cage (you'd weed it down to one after they're established). The cage might only need to be a foot or two in diameter because you are training the plant mostly around the outside, or at least, the blossoming stems. You'll have to keep a close eye on it the whole season to make sure it doesn't set fruit on the inside, since you'd run into that problem you mentioned above - fruit half in, half out of the mesh. Height of the cage I think would be decided by diameter - wider the diameter, the higher the cage. Oh, and I'd suggest careful pruning, too, as the season goes on to keep things manageable and to make sure it's balanced so it doesn't topple from too much weight on one side.
I should add that you may want to make slings for the fruit as it gets bigger - I do this with my heirloom tomatoes to keep them from breaking the stems - some of them can weigh a pound or more! Some of those squash can easily weigh 3-5 pounds.

the season is over but you can research for the next one.
people are using livestock panels or 'cattle panels' to build small arbors. they are very sturdy.
i am sure you can grow small pumpkins on them. and whne they get heavier, you can rig a holding mesh bag for them and attach it to the structure. lots of interesting ideas in 'greenhouses and garden structures forum' on GW

Here is a link that might be useful: cattle panel arbor

I have a buddy that grows butternuts vertically. He nailed some wooden lattice to his garage and lets the vines grow up on them. Seems to hold up pretty well.

I would like some tips on growing snake gourd. What type of soil, seed source and trellis structure. Thsnks