If you can find it ,Isinglass is a wonderful way of preserving eggs.
You will need an enamel (NOT plastic) bucket and isinglass dissolved in water. put eggs in bucket as they are gathered and make sure they are covered in the liquid at all times.They will keep all winter like this and will be good for cakes or scrambling but not so good for fried eggs.
My mother's Springer Spaniel,Jimmy Giblets (dead for forty years now) would hunt out eggs laid in hedgerows and bury them in the veg. garden. Ma,digging them up in the winter would find them good enough to eat so I suppose it's the lack of air that keeps them fresh.