While there are good substitutes for egg whites in cocktails, such as the aquafaba from canned chickpeas, I don't think that will work well in this particular recipe. You need the egg yolk for that eggy taste and the egg white for a bit of foam. It is eggnog after all. I haven't researched or tested egg yolk alternatives that I can recommend, but if you have one, feel free to share it in the comments below.