I was on a gathering quest in the Misty Peaks when I ran into what looked like a giant emu.
I attacked one and, lo and behold, it laid an egg in surprise. I picked it up and, remembering Powderstones and Wyvern Eggs, I thought to carry it back to home base to deliver it to get some more cash. However, when I was near the red box and the icon came up over my head to deliver Account Items, the Egg wasn't delivered. It stayed in my hands no matter how much I mashed the A button.
Best Answer
You can only deliver eggs to the red box when it's a quest objective.
If you get an egg at some other time, you can bring it back to the Veggie Elder at the camp. He'll give you a Jumbo Pearl for a regular egg, and a Wonderful Beak for a golden one.