Thanks, everyone, for the advice. I modified our nesting boxes to be rollaways and that seems to have solved the broken egg problem. She still hangs out in there too much and poops in there more than I'd like, but we've had zero broken eggs since I made the modifications.
No, I see her eating and drinking all the time. I think the biggest problem is that she scratches all the nesting material out whenever she goes into the nesting boxes and breaks eggs that way, which led to her developing a taste for them.
We have a one-year old polish hen who is at the very bottom of the pecking order--I sometimes joke that in our flock of eight she's ranked tenth--who often separates herself from the flock. Lately she's taken to hanging out in the henhouse and nesting boxes. She scratches out the nesting...