One of my dogs swallowed a chicken bone one. Basically, I panicked and took him to the vet for an x-ray. That is exactly what you should do but I will say not to panic. The stomachs and intestines of most animals are extremely thick and the digestive acids are very strong. Most likely the bone will dissolve before he/she passes it.
I am not a doctor nor vet. I am just some smock on the net but I can say that the biggest worry about chicken bones are getting lodged in the throat not when they get swallowed