Answer: Squash really does take a lot of room, as you have discovered. You can train the trailing stems up a trellis by either tying the stems or coaxing the tendrils around the trellis. Since the fruits will be heavy, you'll want to support them in a sling. I've found that lengths of old pantyhose work well because they stretch as the squashes grow. Just make a sling and tie it to your trellis, then gently roll the squash into the sling.
Best wishes with your garden!
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