What ever you grow they will try and eat it.

Maybe you can put an extra layer of chicken wire over it, so the run holds the chickens in, and that extra chicken wire is what the vine grows on? That way they can't really reach and eat it.

How tall is a chicken run? I don't know what to recommend. Hyacinth beans or scarlet runner beans would be really pretty.

I'd like to have a couple of pet chickens, so your post about them damaging the garden is very helpful in making me feel like it's not so bad that I can't have them. :)

Really? You think they will eat everything? Bummer. The run is 6 ft tall, and about 50 ft long, in an L shape around a shed.Maybe I'll just focus on planting perennials around it, say a foot or 2 away from the fence.

You could always plant something vining specifically for them, like beans (any beans you would eat yourself). My hens ignored morning glory vines on their run, but did try to eat a passionflower. I'd suggest using the double layer of chickenwire idea, or use anything that will distance their beaks from the vines. Wood panels work fine too. I did have good results planting ditch lilies (common orange daylilies) along the run, then training peas and beans up string on our side of the run. The daylilies are perfectly edible and probably will be nibbled on, but at least the all-important bases of vines won't be eaten. My hens also ignore ivy, but I don't know that you'd especially want it on the run, I try my best to keep ivy in check here.

Now that the hens are roaming the back freely, we start plants in tomato cages. The cages have to be staked, otherwise the hens topple them over.