Hey again @Crackers -- you can export eggs and certain items to your main inventory by first going into your pet's adventure, taking them to the Entrance Gate at Gateway City, and clicking "Finish Adventure." Then, on the home page of the adventure, click the inventory of the pet who has the eggs, and select the egg you want to export. Then click "Send to Main Inventory" and hit "Send."

Please let me know if anything I said was confusing or if you have more questions! ^^