Hi
You say you feed equal amounts of corn...what do you mean by that?
If they are getting as much corn each day as layer pellets, that may well be your problem as they will only be eating half as much layer pellets per day as they should because they are filling up on corn which they prefer. The layer feed has calcium added to it, the corn does not, so they will be taking in just over half as much calcium (corn will contain some calcium but not the enriched level in layer feed) each day than if they were only getting the layer feed which is designed to give them everything they need to be healthy and productive. They will also be getting less protein because the corn contains significantly less than the 16% protein in the layer feed.
If one of your most productive girls is one of the ones having problems it is most likely because she needs more calcium and protein than the rest as she is producing more eggs. They store calcium in their bones, and use that to top up when they are low, but eventually that deposit of calcium will run low if it isn't getting topped up properly each day and then problems can start to arise like poor shell quality, egg binding and prolapse.
Corn is also inclined to make them fat, even obese. I butchered some unproductive hens for a neighbour who fed half and half corn and layer pellets...the layer pellets were available all day ad lib but the hens only ate roughly the same amount as the corn because he threw a scoop of corn down for them each day and put a scoop of pellets in the feeder to keep it topped up. Anyway, those hens had yellow fat deposits 2inches thick on their abdomen and most of their organs (heart, liver, gizzard and intestines were encased in fatty deposits to the point that it was limiting function.... and probably why they were hardly laying any eggs. I have some photos somewhere of their insides and the shocking amount of fat in there.
Anyway, moral of the story is that too much corn or other high carbohydrate treats like bread, rice and pasta is bad for them. Limit it to less than 10 % of their daily intake or cut it out altogether for a while and see what happens. If you really want to continue feeding corn then make sure they have access to a source of calcium like crushed oyster shell in a separate dish, not mixed into their feed as too much is not good either and they will know when they need it. You can also dry and crush their egg shells and feed them back to them.
My layer flock get a cupped handful of fermented mixed corn each day which is probably half as much as if it was dry because it swells and that is between 14 birds and the rest of their diet is layer pellets. They probably go through about 2 lbs of layer feed a day. Hopefully that gives you an idea of the level of corn you can treat them to without affecting their health and production.
Best wishes
Barbara