Most small birds are afraid of cats, rats, hawks, owls and snakes. I've read that when they travel in flocks (with other species like chickadees), they may attack a predator. So I'm not sure the birds would be deterred if you painted those designs on the wood. My sister had woodpeckers drilling her shed. She tried the plastic owl and kept moving it around, but after a while, the birds weren't afraid of it. She doesn't have chickens. My guess would be it might frighten your flock.
I've read that if you can record the woodpecker's distress signal and play it in the area, it may keep them away.
Remember though that woodpeckers (like your chickens) are omnivores, So they are great predators for ants, beetles, caterpillars, plant lice, scale insects, spiders, weevils. You might look to see what insects are making homes in the wood. If you can rid the wood of the insects, the woodpeckers will move on.
Good luck!
regards,
keljonma
if edited, probably for typos...