I'm sure half the posts here start with this sentence, but: I was reading a wikipedia article and now I have more questions than answers. Since there's no /r/askfolklorists, this is my best bet.
The article in question is changeling, and it mentions eggshells a lot. To quote the relevent passages and their cited articles:
Several methods are known in Germany to identify a changeling and to return the replaced real child:
confusing the changeling by cooking or brewing in eggshells. This will force the changeling to speak, claiming its real age, revealing its position beyond synchronicity.
Sadly, I cannot find the cited passage. Going to page 364 of the English version gets me nothing, a CNTRL+F for "eggshell" or "eggs" gets me nothing, and a CNTRL+F for "egg" just gets me lots of Scunthorpes that don't mention actual eggs or eggshells.
The tale of surprising a changeling into speech – by brewing eggshells – is also told in Ireland, as in Wales.
A search for "Yeats changeling" yeilds a poem I'm very familiar with - The Stolen Child, but is definitely not what I'm looking for, as it's a century too old. I did manage to find Yeats The Initiate: Essays on Certain Themes in the Work of W.B. Yeats, first published in 1986 - this could be the book I'm looking for, but using google books, I can find no references to eggshells and only five references to eggs - all the symbolic "world-egg".
The listed pages do mention faeries (page 49), but no eggs, and page 48 wouldn't render.
Now, brewery of eggshells I can get half an answer for. Unhappily, many search results are just for hundreds of websites quoting the exact same story, so I don't know if this was really widespread or if there was just this one story. Here is one such website - the one the wildly insufficient "Brewery of Eggshells" wikipedia page cites.
So when the day of the reap came the woman did all that the Wise Man ordered, and put the eggshell on the fire and took it off and carried it to the door, and there she stood and listened. Then she heard one of the children say to the other:
Acorn before oak I knew,
An egg before a hen,
But I never heard of an eggshell brew
A dinner for harvest men.
This is the version of the story that I know. I also managed to find an Irish variant:
Mrs. Sullivan was silent in expectation, and Ellen continued, “Put down the big pot, full of water, on the fire, and make it boil like mad; then get a dozen new laid eggs, break them, and keep the shells, but throw away the rest. When that is done, put the shells in the pot of boiling water, and you will soon know whether it is your own boy or a fairy. If you find that it is a fairy, take the red hot poker and cram it down his ugly throat.”
[...]
“Oh!” shrieked the imp, standing up in the cradle, and clapping his hands together, “I’m fifteen hundred years old, and I never saw a brewery of egg-shells before!”
All I can find are those two stories repeated, verbatim. No commentaries.
In Poland, apparently:
Still, even if a child was taken by the Mamuna, there was a way to have her return the baby. The mother would take the changeling child to a midden, whip it with a birch stick, and pour water from an eggshell over it, all while shouting "Take yours; give mine back." Typically, the Mamuna would feel sorry for its own child and would return the human baby to its mother.
In Spain:
The legend says that in order to distinguish a "xanino" from a human baby, some pots and egg shells should be put close to the fireplace; a xanino would say: "I was born one hundred years ago, and since then I have not seen so many egg shells near the fire!".
I can't seem to find any discussion as to where, when, or how this connection between the fae, or more specifically changelings, and eggshells came to be - but most cultures with changeling lore also utilise eggshells in the return of the human child. Is it possible that an explanation has been (indirectly) documented sufficiently to be divined, or is it simply lost to the passage of time?
I do note that cuckoos (and other birds with this behaviour) are mentioned at the bottom of the article, so it's conceivable that that's what gave rise to the eggshell association, but it could also just be a coincidence. I don't even know if it's possible to answer this question, but to reiterate it for clarity:
Where did the association between changelings/fey children and eggshells come from?
You're probably placing too much significance in the very specific eggshell motif. There are other, similar motifs that appear in this legend type: some describe the attempt to lure the changeling into speech by cooking porridge in a pot jammed full of spoons, but chasing a relationship between fairies and spoons will lead to nowhere.
The changeling legend was classified by Reider Th. Christiansen as Migratory Legend 5085. The "migratory legend" part of this indicates that it is a story that was/is told to be believed, that it travels, and that it holds together wherever/whenever it travels with a fairly consistent set of motifs.
The basic plot of ML 5085 is that a woman has an infant who seems to be failing to thrive, even though he (it is usually a boy) had spent his first few months growing and maturing at reasonable rates. The mother consults some of the old women in the village, and they explain that the infant has likely been exchanged by the fairies: the real baby is in their custody and they have left one of their own who is enjoying the hospitality of the human household. The old women recommend a cure. At this point, the legend follows one of several different approaches, but the ending is fairly consistent, namely that a fairy woman appears with the real infant and takes the changeling back to the realm of the fae.
The "several different approaches" indicated above can be grouped into three generic motif patterns: the mother abuses the changeling until the fairy woman appears with her real infant; the mother takes the infant to church to be baptized or in some other way involves a church, inspiring the changeling to speak (because he objects to being involved in Christian ritual); or the woman does something remarkable in front of the changeling to coerce him to speak, revealing his true character.
This third option is, of course, the one that involves the eggshell motif, but as indicated, this is merely one of several options intended to surprise the changeling, who speaks and reveals his nature.
The changeling legend is extremely old: there is a hint of it in Petronius Arbiter's Satyricon of the first century CE. There are many latter references to the migratory legend, including one in the writings of Martin Luther, and of course, the legend appears in the folklore collections of diverse European nations. In short, the legend type has been around for a long time, it is widely distributed in Europe, and it has had a lot of time to develop and settle into various forms. One of these forms, as you have found, includes the idea of brewing or cooking in an eggshell. The point of this motif is to employ a tiny vessel to achieve something that is usually done in a large pot. The eggshell serves as the "tiniest of vessels" that a farm would have on hand. It has no more significance than its function - as far as I can see. We are not seeing some underlying connection between fairies and eggshells. It is simply a plot device.
I will follow this with an excerpt from a book of mine. Let me know if this helps or if you have questions.
edited to correct a date!