Thanks everyone for all the responses. I'll definitely be looking into these options.
The distance isn't far, maximum 12 meters. I like the idea of having a webpage to show results - @wildbill, what do you think of @JBBOP's idea of using another ESP (instead of a PI) as a master? I'll check into MySQL and MaiaDB, thanks for the recommendations.
Thanks! Any advantages over a regular SD card for my application?
Yes, the correlation of hen to egg is just the last tag that was read when an egg is detected. A hen can go in and out of a box many times before laying an egg, and this takes care of that too. It doesn't matter when a hen leaves the box so that isn't recorded.
That's an interesting point, I hadn't actually thought about using one transmitter for several boxes. My initial thought was to handle hen ID after transmission, an example ID would be BO-24 or CL-49. If I use ESP, the box ID would be the same as board ID. I'm not sure if it's possible to have several boxes use one controller, because how would eggs from separate boxes be differentiated? I think I would only be able to tell that within the group, a certain hen laid an egg. (But maybe there's a way I'm not seeing here).
A timestamp would be nice, maybe add an RTC? It's a feature I've thought about before.
Nice, that eliminates a lot of memory storage devices or work getting it into EEPROM. Thanks for the suggestion.