Our broodies will occasionally eat a "suspect" egg and, the following morning, will squirt out a yellowish white liquid that, on the sand of the run, looks like some modern art masterpiece. More often the broodies break a "bad" egg just enough to carry it away from the nest (to a distance of 20-30yds). How they identify a "bad' egg? By a process not well understoodMy turkeys have never eaten their own eggs, not once, until I crack them for them, so this is what I do, I also do it with my duck eggs and dirty chicken eggs, not saying you should do it, but this is my experience, and I don't have trouble with it, of course my birds are allowed to forage and do natural things, so they aren't bored or have a need to develop habits that are from improper confinement. It's not natural for birds to crack open and eat their own egg, but if one gets cracked it's fair game
Slate hen caught dropping bad egg far from nest (seem almost depressed about it - reason I caught the shot was that she seemed to be "slouching" as she walked around house and I first thought "injury"/"predator").
Royal jenny taking a malformed egg, away from nest: