Egg nog

The reason you won't find 16th century recipes for "egg nog" is the term didn't appear in print until the next century. Food historians/period recipes confirm English recipes for posset (esp. sack posset) were very similar to later egg nog. References to 16th century Jamestown egg nog were published after the from 18th century forwards, it is most likely the author was using a newer/more popular & accepted American term to denote an old traditional English holiday beverage.

How old is egg nog?

"By the mid-1760s patrons were drinking eggnog, juleps, sling and sanger in addition to the punch and toddy already available." 

---"Taverns and Tavern Culture in the Southern Colonial Frontier: Rowan County, North Carolina, 1753-1776," Daniel B. Thorpe, Journal of Southern History, Vol 4, no. 2, November 1996 

[1796] "Rich and creamy dessert drinks, such as eggnog and syllabub, reflect the English heritage in America, especially in the South. In England posset was a hot drink in which the white and yolk of eggs were whipped with ale, cider, or wine. Americans adapted English recipes to produce a variety of milk-based drinks that combined rum, brandy, or whiskey with cream. The first written reference to eggnog was an account of a February 1796 breakfast at the City Tavern in Philadelphia. Beginning in 1839 American cookbooks included recipes for cold eggnogs of cream, sugar, and eggs combined with brandy, rum, bourbon, or sherry, sprinkled with nutmeg. Southerners enjoyed a mix of peach brandy, rum, and whiskey."