Vegetables almost never make hens sick. Only eating all of one thing can be a problem, but if they have access to other greens, and such, they'll be absolutely fine. Once in awhile, if they gorge on something like watermelon, they'll have runny poo, but that does no harm. The real question is, do you want to share your produce with the chickens!
I occasionally let my 8 RIRs free range in my garden to peck at the weeds & grass. I have tomatoes, cucumbers, corn, watermelon, peppers, summer & winter squash, green beans growing. The only thing they really seem interested in is eating the young corn. They nay peck at the other leaves, but really leave them alone. Of course, the tomatoes aren't ripe yet, and they can't reach the ripe cucumbers, but I do have to guard the corn.
They have never shown any interest in the cucumber leaves. But if that was the only thing green for them to eat, they might eat it.
Not sure about the leaves, but mine sure LOVE the cukes! I plant extras just to let them grow big and seedy just the way they like them. Pop them in the fridge overnight and then cut them down the middle. It makes a great, cool treat when it's hot out. Mine will devour the seeds and finally come back and eat the fleshy part, only when they figure there are no more seeds anywhere to be found, silly birdies!