I know this question is months old, but I just finished reading vol. 1 and this thread, and I loved both. Almost every question I had was answered here. This is a question, however, that I believe I can answer. I’m not that fluent in Japanese but I do have a lot of experience with Japanese children’s songs, and I believe よつば is making up her own song about the free egg probably using the melody and part of a line from どんぐりころころ, which is a super popular kid’s song.
Here’s a link to the song: https://www.youtube.com/watch?v=Y3cPq5vmXng
and the line early in the song about an acorn’s adventures is “おいけにはまって…さあたいへん！”