egge being derived from Norse, while eye/eai/etc from Anglo-Saxon æg. The final g eventually came to be pronounced like a y in that environment, a development that apparently did not affect the Old Norse word -- probably due to being borrowed after the sound shift had already occurred, though the doubled gg could have prevented it anyways (not sure about this latter part).

From https://www.etymonline.com/search?q=egg:

"""

This Norse-derived northern word vied in Middle English with native cognates eye, eai, from Old English æg, until finally displacing the others after c. 1500. Caxton (15c.) writes of a merchant (probably a north-country man) in a public house on the Thames who asked for eggs:

And the goode wyf answerde, that she coude speke no frenshe. And the marchaunt was angry, for he also coude speke no frenshe, but wolde have hadde egges, and she understode hym not.

"""

Very interesting! The Dutch for eggs is "eiren" (plural of "ei" pronounced "ey")

Nitpicking, the plural is eieren, with three distinct syllables :) IIRC, there used to both words with -er and -en as plural, but over time the norm became -en so it was tacked onto the -er. Like kind/kinderen. In German Kinder and Eier are still the plural.

Regional differences.