No, they don't cause the salmonella. It's passed in wild bird feces. It's very, very common for chickens to eat mice. I've never had a chicken become sick as a result (nor does my cat). Mice have fleas, but whatever. More protein. =)
Keep their feed locked up tight in a galvanized can and if it's feasible, feed only what they can finish each day so there is no food in the house. No food, no reason for the mice to be in there.
Try securing a little chicken wire cage into the corner of their house that only the mice can...