Yes definately make sure you save the e-mail where she is bribing you for her good feedback...
As a seller I'm not sure I would give bad feedback to this customer even if I could, she's just one of those customers that come around every so often, and you have to deal with it.. honestly I woud be upset too, but to demand a refund and bribe for feedback is not something I would do...
Maybe just send her new eggs, letting her know that if this time they are damaged again, too bad.. in a nicer way of course...