Oh gee, have you tried supplements... she sounds deficient in something, I would give her oyster shells, laying feed, vities & chopped up hard boiled eggs. If you have tried the feed thing & that has not worked she has developed an obviously bad habit (birdy crazy)... so she can not be a keeper for eggs... but I have a few ideas on how to get her eggs before kicking her out of the flock.
Before getting rid of her I would isolate her and put a wireless camera in the pen hooked up with your computer to monitor her, you need to determine when she is likely to lay, and then you have a good chance at timing when to be up and about to save those eggs. You could also modify the nesting box she uses so the egg rolls away into an area she can not get at, like the commercial layer operations. There are no guarantees with either of these ideas (I never had a hen eat her own eggs) but I think they could work. You could also try "breaking" her of peking her eggs by putting fake eggs that are hard in her box (though with bad habits that will be a long shot and take probably a long time and you have to be there to get her eggs so she gets no positive reward for peking her eggs only the negative reward of nothing from the fake eggs).