See also Preserving the Harvest - Fresh Eggs: How to preserve eggs (at the Hive blog) for an updated version of this article.
Last year and this year, at the Hartwell Tavern food preservation event, I put up fresh eggs, in an 18th century manner, to last all winter without refrigeration. This is the recipe I was given, but I don't know where it comes from:
To use lard or shortening to coat the eggs, first melt the grease and cool it till it begins to solidify again. Dip each egg in the melted grease individually and set them on a paper towel to dry. When the shortening or lard is dry on the eggs, rub the eggs with a clean towel, removing excess solid grease. Rub gently and buff each egg. Now repeat the process, before the shortening solidifies. Work fast, allowing the shortening to get almost solid before re-heating it. When dry put them in a box of ash and keep them covered until use.
Eggs should always be packed with the small end down, because the yolk will not settle toward the small end so readily as toward the large end or the side. It is important to have clean eggs but do not wash them as it removes the protective covering.
I've since done my own research; see below for documentation.
Last year, I melted commercial lard (the kind you can buy in some grocery stores, that's labeled "lard" and "manteca"). It was a chilly day, so I hardly had to let it cool at all. I rubbed the eggs, and then rubbed them clean. I had ashes sifted from the fireplace and a wooden box, and I put in a layer of ashes, set in eggs small end down, covered them with ashes, laid in a second layer, and covered them with ashes. I put the box in my "root cellar" (my basement) and kept them there from October. Every week or two I went down and fished out another egg for breakfast, and each egg was as fresh as the last. In mid-April, when my basement became palpably warmer, I put the the last four eggs in the fridge, and finally used them to make Hannah Glasse's recipe for nun's cake for a tea at the Needham reenactment, which cake was eaten with relish by all (particularly as I had substituted currants for caraway seeds).
Why ashes? Why not sand, as you use for potatoes? It might be for some chemical property of the ashes, but after fishing out eggs all winter, I think it's because ashes are soft, and if you were scrabbling your fingers through sand, you'd be too likely to break an egg. The Universal Cook, and City and Country Housekeeper recommends salt, but essentially everyone in the 18th century had a fireplace, so ashes were free for the labor of sifting them.
Why small end down? I am told that it is to keep the yolk from settling against the shell, but I haven't found documentation of it.
This year, I used a mixture of leftover bacon fat and sausage grease. It was messier, but we'll see if it works as well. Also, I buried the eggs in last year's ashes, and I found I couldn't pack the eggs as closely. It would have been better to empty the ashes into another container and layer them back in as I went.
Here are some more experiments to try:
From The Universal Cook, and City and Country Housekeeper, by Francis Collingwood and John Woollams, 1792, p. 333.
From The London Magazine or The Gentleman's Monthly Intelligencer, Vol. XIX For the Year MDCCL (1750), p. 180.