The very first scene starts off with Robert de Baudricourt brow beating his poor Steward because there aren't any eggs. We can understand. We also get grumpy when there's no breakfast. The hens haven't been laying ever since Robert refused to see Joan. The scene ends with Robert giving in and supplying Joan with the soldiers and supplies she needs to go see the Dauphin. Immediately, the hens start laying again. Is this just complete randomness on Shaw's part or is it incredibly symbolic? Let us examine:
Eggs are an ancient symbol of birth and renewal. It seems pretty logical, right? Eggs=little baby birds=birth. This symbolism goes all the way back before the Romans and Greeks.
It's so old that nobody even knows where it came from. Ever heard of an Easter egg? Ever wondered what they have to do with Jesus being resurrected? It comes from when the Catholic Church was going around converting all the pagans. "Pagan" is the term Catholics made up for any and all of the indigenous religions they came into contact with. Pagan basically meant you weren't Catholic. Anyway, the pagans had lots of holiday traditions that they were pretty reluctant to give up – one of which was celebrating the spring equinox with eggs. Get it? Eggs=little baby birds=birth=spring.
The Catholic priests found that it was a lot easier to convert people if they let them hold on to a few of their old traditions. They were like: "OK, you can still have your spring festival and your symbolic eggs, but we're going to celebrate Jesus' resurrection now instead of the renewal of the earth." (Incidentally the word Easter comes from the Saxon goddess Eastre.) The pagans were cool with this switcharoo. So, now we have: eggs=little baby birds=birth=Jesus' resurrection. (You can learn more about Easter here.)
When Robert decides to help Joan begin her quest, it's a birth of a sort. Only, instead of a fuzzy little chick, a saint is born. The symbolism is doubly meaningful because Joan is a Christ figure. She ends up being martyred just like Jesus and even resurrected in a way when she is made into a saint. OK, so the final equation is: eggs=little baby birds=birth=Jesus' resurrection=Saint Joan of Arc.