I am also in camp layer feed. This way, your hens will be forced to consume extra calcium, not pick at it as they please and potentially become deficient. You can still keep the oyster shell stations out if you want.
And also because it seems to me that the added calcium MAY do the roosters...
I wonder if she just ate something weird, that caused that to come out. I also wonder if it was a piece of shed reproductive tract, as mentioned above by someone else. I am actually leaning towards it not being egg material, myself. But, who knows for sure