This is something I've often wondered about.
For one thing, imagine all the devices that would have been invented over the ages to protect the egg, and to carry the egg, and to keep the egg warm. There would be a new group of crimes much like child kidnapping, in this case eggnapping.
Just as we keep a few locks of hair from the baby's first haircut, we would also keep pieces of the baby's eggshell. We might have two birthdays, the dai Mom laid our egg, and the day we hatched. Instead of wishing someone a happy birthday we would wish them a happy hatchday.
"Quick, Harry, grab the camera the baby's hatching!"
On a more serious note I have wondered, if humans laid eggs would they have a hard shell like bird eggs or a soft leathery shell like most reptile eggs? I understand the two egg laying mammals the platypus and the echidnea lay a soft leathery shelled egg more like reptiles.
Human civilization would probably have become sentient and developed villages, towns, and cities much earlier than in OTL. In fact humans might not have gone through a long nomadic stage at all due to the need to carry or transport unhatched eggs.
Or perhaps women would have laid their egg each year only in one or two seasons. Thus a nomadic or semi-nomadic tribe would have only traveled during those times of the year when there were no unhatched eggs to transport.