The only opening is an automatic door, chicken-sized where they walk up a ramp and enter. There's a red light flashing constantly throughout day and night on the door. I even put a fake owl up on the gate. I've even stood there watching it enter my chicken house, asking it what it's doing. It lands on the ground 10 ft from me and looks at me. The jays do what they want. I have now lost 14 eggs in 5 days. The ONLY answer in my instance seems to be to remove the offenders before they teach my hens to eat their own eggs! My nesting boxes are integrated into the building so I'm not sure I could add a rollaway feature easily, but it also sounds like hens don't like them!I would be worried that the jay would bring her/his fledglings to the coop and you'd have a gang to worry about.
How about a cover over the run?
How about hanging strips of cloth or plastic over the door? The chickens would learn to walk through it, but the flying jays are unlikely to figure that out.
How about getting an electric light that looks like an eye. It blinks on and off. Supposed to look like predator eye. Put it just above the door.
Last year I tried closing their automatic door and opening the side gate so they all have to go around the back way, but all my babies are in there now so I can't do that for at least a few weeks. Pretty sure I lost eggs even then though. They're impressive.