I haven't looked at these for quite a while, but if they still work they may be of some benefit.
Virginia Tech – Stopped Laying
http://pubs.ext.vt.edu/2902/2902-1097/2902-1097.pdf
Florida – Stopped Laying –
http://edis.ifas.ufl.edu/pdffiles/ps/ps02900.pdf
When hens stop laying, the molt is a huge possibility, especially this time of year. That's my first thought, mainly because of the time of year.
Next is that they are hiding a nest on you. That can happen any time of year. With that egg disappearing under the broody, this is not likely.
Something could be getting the eggs. It could be a chicken eating them. They don't always leave residue but often they do. If it has been going on a while, that's probably not it. I had one that would purposely open eggs to eat them. But she would only open one or two a day and she often left traces. The other hens would help her eat the opened egg, but they never learned to open one themselves.
It could be a snake. I've had a snake eat the eggs out from under a broody before. But usually a snake will not come every day. It eats the eggs and goes off to digest them, only showing up once every three or four days. With them all 100% disappearing, I seriously doubt it is a snake.
It could be a lot of other things. Some animals like skunks and possums prefer eggs to eating the chickens, but they will take chickens too. Others will eat eggs but usually prefer chickens. Don't believe that stuff about predators only coming out at night. Practically any of them will hunt and feed during the day. But usually with these animals they don't necessarily come every day. Anything is possible but I'd expect you to at least see an occasional egg if it were an animal. Usually the ones that prefer chickens will carry the eggs away so you don't see a trace. The ones that prefer the eggs usually eat them on the spot and leave evidence. Not all predators read the book and act the way we think they are supposed to, so it could be any of these. But they are not real high on my list of suspects.
Do you have a family dog that has access? Has Rover learned that the egg song is an invitation to a snack? If you have a dog that has access, she would be pretty high on my list of possibilities.
With all eggs disappearing without a trace, a human would enter my thoughts too.
I really think it is a molt but I'm not looking at them and you are. Hopefully this will help you some.
Good luck. These things are not always easy.