It sounds like someone ate the eggshell and left you the sticky mess
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Depends, really. As a general rule, an egg is broken and the hen/pullet instinctively eats it as a way to deter predators. They can break for a lot of reasons, whether there isn't enough nesting material in the box, thin shells, or as Wyo said she may have been rearranging and broke it.
I also think one will eat an egg if they're trying to get some missing nutrients. If you've got them on good feed with a side of oyster shell, than it was probably an accident that will happen again. In my experience, it's inevitable
Unless there is something else going on, I don't think you've got anything to worry about.
Our vets here don’t see chickens. Called many yesterday for the fecal float test. Plus. What chicken would I takeMaybe get them on some oyster shell to strengthen the shell. But you may need to bring in a vet.