Eating greens really isn't that important to chickens. They will do it, certainly! However, dry matter (aka seeds, grains, even dried hay) is where they receive alot of their nutrients from. They don't really have the proper system to take nutrients out of fresh greens (think of how a ruminent like a cow needs so many stomachs to get the job done, your chickens are literally only working with a gizzard full of rocks)
That is not to say that I throw the idea of greens out the window. We have found that weeds and clover (which is protein rich) are a huge hit with most chickens (not just ours, I'm a nerd and subscribe to printed newsletters...yepp I really do!) Anything you can plant that will go to seed quickly is also good. I have heard of people leaving their gardens after the season is over and when all of the plants go to seed they let their chickens loose for a grand time!
Plants that attract bugs are good for chickens, we also let ours into the compost pile and they will turn it for us while searching for bugs.
But you cant expect to let a plant grow under a chickens watchful eye. If they don't find the seed...well they do love fresh sprouts they are just so soft and tasty! Chickens will never just let something grow, they are too curious and will surely get into it before it's time.
I do agree with all peeped out, that things that are still planted seem to be easier for them, only our geese will eat greens that we just throw in the run.
We have had a bit of a heat front and have been chopping and freezing large chunks of squash, zucchini, cucumber, tomatoes, whatever the garden has that isn't quite fit for human consumption because of little bug bites or growing too big. They mostly seem to enjoy these treats! You could also offer seed pod veggies like peas, green beans, black beans. Picking them and letting them dry a little bit is even better. Just not soy beans unless you roast them first. Whole corn cobs are fun (but ummm they seriously come back out looking like they went in if you know what I mean...)
Hope that gives you a few ideas for veggies that aren't leafy.