Do you have enough nesting boxes with golfballs or fake eggs in them? You only need one golfball or egg in each nest to make the point. She shouldn't be dropping them from the roost, it sounds like there are not any nests available or unoccupied.
It is natural for the other chickens to eat broken eggs, mine do it like they're starving. It is an unwanted behavior for the chickens to "break" the egg before eating, because they'll start eating eggs from the nest. A nasty and annoying habit.
Make sure you have enough nest boxes with a golfball in them for your hens, before thinking it's a health issue. I think it has to do with a nest box issue, myself. Maybe there are not enough to share and she doesn't have the rank to fight for one.
I had a hen that did that for a few days. Some times their clock gets messed up. Is she laying during the night? If it's dark, she can't get down to get to the box. It should work itself out. Is it possible to have a low watt light bulb on so she can see?
I have 5 boxes and she is the only one old enough right now to lay, she had been laying happily in one or the other of them for the past year, until lately....
I will do the light idea tonight. great idea. will also put an egg in one of the other nests to re-mind her....
My gut feeling is that she's lazy, cold, ect and not inclined to come off her roost when the urge hits.
another question - she's been dropping off her roost for a couple weeks - shouldn't her clock have ticked to a later time in the day by now?
1 you have enough nestboxes.thats not the problem.she has gotten used to laying eggs while roosting.been there had that problem.you need to catch that hen an put her in a pen.but do not give her a roost.make her sleep on the pen floor.an sooner or later when she lays her egg.it to will be on the floor.you can put a nest in with her if you want.but leave her penned for 3wks to a month till she learns to lay egg on pen floor or nest box.