Chickens, at least adults, will kill mice and younger rats, but only during the day. At night, rodents are safe.
Rats will eat eggs and kill chickens, and must be eliminated however you can. Mice are more of an ongoing problem, must keep them down to near zero.
Management! Always a...
I'm not out there at dawn, so feed and water are available 24/7 at the coop. And we feed an all-flock feed that's fine for all the birds, laying eggs or not.
Mary