My finch recently was egg bound, with a soft shell-less egg.
I would advise picking her up and feeling around her vent, see if you can help her push it out by massaging around that area
Button quail are naturally more flighty, making them harder to tame, but it IS possible.
Try putting your hand in the pen often, offering them food, and even holding them.
My quails quite like to be gently stroked along their back :)
Good luck with your buttons!