Good Evening
I raise Button Quails in Tennessee,have experience with them back from Europe and have similar experience withthe problem also in the birds that I raise.( Finches,canaries,budgies,cockatiels)
In this situation withthe Button Quails I would rather offer bird feed to lower down protein level that would make them not laying every day ( works with mine ),so you may get egg for a two-three days.
The problem that you have is common if females are to young.Not all the chicks develope in the same time. With mine Button Quails I am waiting for 10-12 weeks of age before I would like them to start laying.
Eggbound in birds happends if I could say for two reasons,or lack of Calcium in diet and female body cant complete hard shell,so egg is like e yello.Second reason would be diet thats a constantly dry,in this case you should offer partial seed diet coated with oil ,or simply rape seed and sunflower seed chopped up.This would lube her female parts.
I offer my Quails Custom mineral mix in which one I include oyster shell,sand,charcoal ...Most of the people think if they offer grit everything is fine,but if you dont have propper level of phosforus ,or D3 calcium wont be absorbed.If you could get hand on the fine bird grit, pigeon gritt would work better if you could grind it down.Liquid bird Calcium enriched with D3 works good as well.
When females cant pass the egg due to egg size sometimes it takes them more then one day,they dont eat properly like they may did past week ,so they do lose energie and extra spot light with heat works very good to help them recover.I do also offer Electrolytes in the water.
Good luck TNQuail....