So watching the first episode of season 2 of TURN. Why does the spy, Culpepper, need to write on eggs? He has a little hidden notebook and various other hiding spots but wants to use eggs? The whole egg thing makes no sense.
The main reason he's using eggs is because he knows he and his luggage will be searched every time he enters/leaves New York. Given that the messages on the egg are underneath the shell and not visible, even if they're found, they're less likely to raise suspicions than a notebook or letters--particularly if said documents are in cipher.
I also think that this is, in part, either foreshadowing, or a play on how John André was historically captured--because of documents hidden in his stockings.
Also, you may want to consider that this is a spy show run and written by the same folks who did Nikita, not a show famed for its incisive story logic and sticking to factual elements. :) And they probably really really liked calling back to them beginning to use the eggs in Season 1, (episode 6 "Mr. Culpeper").
I haven't actually seen this show yet. However, there is a method by which you mix alum and vinegar and use that as 'ink' to write on an egg. Once you've boiled and peeled the egg, the message appears on egg white. Supposedly this was developed during the Spanish Inquisition by Giambattista della Porta.