I have depression, anxiety, and migraines, and I don't have any chickens right now, but I find the rabbits, horses, ducks, geese, and turkeys to be calming. The turkeys play by following me around until I turn back and look at them. If I wake up with a migraine, my mom let's them out in the morning, and they hang around the house churping and purting until I come out to see them. They climb the stairs, sit (And poop) on the barbecue, perch on railings, and hang around my window, since I've got a wild bird feeder. They only really tolerate being petted when they're on the roost right now, but they seem to prefer me to their geese buddies.
Baby rabbits are just heartwarming balls of fluff. If you pet them regularly, they look for you when you open the barn door. We've got a hinge-top growing box for our litters, and the 4months that are in it now all climb up for pets. Horses, if you have the time and money, tend to understand depressed humans best. They can usually pick up on emotions and get when to be cuddly. I've got a 5 year old gelding who we bottle fed, and he's more in touch with how to be people than horse!