Jays pecking eggs

[canman47]

I have falsely accused my hens of egg eating, I put a camera in the nest area to see who was doing the damage and to my surprise a Scrub Jay flies in frequently to peck and eat eggs. Also an egg got left in at night and a rat carried off a whole egg. Don't know what I can do. The Jay has to go through a tunnel then into the main coop then into the nesting coop and then fly out again which it is able to do in a few seconds. The rat seems to make a nightly visit about 2am but usually there is nothing there. There are 2 types of baited traps as well as a bait station but the rat shows no interest in these. Pretty hard to keep rats out. How bad are they? We live in Walnut orchards so there's always plenty of food for rodents plus there's no way to avoid food litter on the ground.