The hen laid an egg, it wasn’t ordinary – it was gold. (нести яйца = lay eggs)

A mouse runs past, its tail touches it, the egg falls down and breaks.
Another version is "Бежала мимо мышка, хвостиком махнула (waved, swung, swept), яичко упало и разбилось".

яичко is a diminutive for яйцо so it's not like just an egg, it's more like a "little egg".

will bear you an egg
not sure bear is the right verb. снесу here is the same as нести (lay an egg).
It's something like "I'll produce a new egg for you, not gold, but an ordinary one"