Check out caigua its a great cucumber like crop which grows very well vertical.

Don't forget peas and stringbeans!

Cucumber - Poona Kheera. I found that it trellised nicely and was resistant to just about everything, including drought. They do, however, look like a potato (seriously). It doesn't affect the taste.

Summer Squash - Tatume and Tromboncino. Both still do well later as the days grow cooler and don't seem to be bothered by powdery mildew in my experience. Great on a trellis, but Tatume will take off if not trained. Tatume tastes just like zucchini, Tromboncino is a bit different.

Winter Squash - I trellis them all, but a 30 pound squash is a challenge at the top of a 7 foot trellis. I use slings. Black Futsu and Kikuza seem to be OK without slings on the squash. They are also squash vine borer resistant.

Kikuza Squash:
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Bellatrix

For large lima beans, Christmas and King of the Garden are both good varieties.

I'll second the recommendation for "Tromboncino" as a summer squash. It is an aggressive climber, and will easily climb to the top of virtually any trellis - and beyond! the long zucchini-like fruits (up to 24" long in harvest stage) grow straight & beautiful on a trellis. I say "zucchini like" because Tromboncino is a different species than regular zucchini, with a slightly different taste & texture; but it can substitute for zucchini in any recipe. Since the fruit is seedless for 90% of its length, it is outstanding for freezing. Very resistant to insects & disease, which is the reason it has permanently replaced zucchini in my garden.

For winter squash, you would want a variety with strong tendrils, and aggressive vines. "Queensland Blue" (C. maxima) would climb well, but like Bella Trix mentioned, the 20 pound squash would need additional support as they gained weight. The varieties she mentioned are also good, and might even do better in a hot climate; but it's worth noting for seed saving purposes that they are C. moschata, the same species as Tromboncino.

I trellis all of my cucumbers, but in my experience, some of the Asian varieties are the best climbers. Not many will climb to 6 feet, though; 4-5 feet is more realistic.

My guess is that you did not request information on pole beans because you've already figured out that they can climb. ;-) But I could recommend some good varieties if you are interested.

I grew butternut squash this year on a trellis. It did allright but I think maybe growing on the ground they root more and you may get more fruit that way. I'm going to try a row of short trellis so they will grow up a little then back on the ground for rooting. They were alot cleaner on the trellis and better to store.

No idea if black passionfruit will grow in your area, but my vine also serves as a handy bean trellis.
Rocoto chillies are lanky, climby plants. They're also a highland chilli which is perennial in my climate.

Dennis, I don't know about the OP, but I would love to see your recommendations for pole beans. I have only tried 3 and would like to experiment more.