My cousin was wondering about saving eggs without refrigeration the other day. She lives in a very rural area, and if there was an ice storm or damaging wind storm, there's every possibility that she might be without power for some time.
Imstillworkin, on youtube.com has been conducting an experiment comparing different ways of storing eggs. She's trying refrigeration, non-refrigeration, coating with mineral oil and vacuum sealing. She posted a two-month update the other day, and I've embedded it here for Jill.
If I had more half-gallon jars, vacuum sealing might be a good option. Personally, I'd opt for coating in mineral oil. I've also seen posts about people coating with butter, but I'm leery of the small amount of butter going rancid. Years ago many people used sodium silicate for storing eggs; this was called waterglass. They'd mix a solution in a big crock and immerse the eggs in there. If one googles "water glassing eggs," enough results show up to satisfy curiosity.
Since mineral oil is easily obtainable, and water glass is something: 1. I don't really know where to buy and 2. don't seem to have many uses for, mineral oil would be my medium of choice. We've enough junk around the house without adding something else that is one-use specific.