Depends on how big the snake is and how big your chickens are. If there chicks then yes. Especially if they are the size of the rats. Rat snakes will also eat eggs.
Rats will bite the heads off smaller chickens as well as nibble on toes and legs of older chickens. They will also eat and carry...
I've had several roosters that didn't crow. If they are secure in there status and never challenged and everyone submits to them and there's no other roosters they didn't have a need to crow. My last on looked like this too.