There is nothing particularly tricky about the word eggnog.
It, like the concoction it designates, is a combination of egg, the product of the hen, and nog, “strong ale,” the product of the brewer.
Originally, that is, strong ale was the spirituous ingredient, but it is often replaced by wine, rum, cider, or other spirits.
Incidentally, the beverage is of American origin, known by this name in Revolutionary times.