I have witnessed egg eating on the coop floor at 5am, when I open the coop and turn on the lights The egg was laid from the roost and broke.
I have also seen a hen eating a thin shelled egg in the nest box. It broke when laid or stepped on. My hens don't eat normal shelled unbroken eggs.
I keep a fake egg in each nest. You say they kick out the fake eggs. I bet they kick out the straw, hay or pine shavings too. The hen stands up when egg is laid and if there is not enough bedding in the nest, the egg can break, and they do check the egg after laying. If it's broken they may eat it.
To keep them from kicking out bedding and fake eggs. Put a 4 to 6 inch barrier at the front opening of nests to help keep in nest material. Refill as needed to maintain at least couple of inches.
. I find straw is the least likely to be kicked out.I check for eggs three times a day, four times in sub freezing temps. GC
