x2 what @cluckcluckluke said.
Some chicken feeds are mostly vegetarian, but they will get better nutrition if they have some animal protein in their diet. It's very difficult to get all the essential amino acids in the right balance from a vegetarian diet.
Unless you lock your chickens up 24/7 in an insect-screened cage, they will be eating a goodly number of insects. Possibly mice and small snakes, as well. I know some people never let their chickens out, but I think they are so much happier to spend at least a little time free-ranging.
Cats are true carnivores...they will get sick on a vegan diet. Chickens can get by on vegan/vegetarian, but they won't thrive as well.