Hey there @Cyrinus -- in order to export eggs and items from the adventure, you'll first have to take your adventuring pet to the Entrance Gate in Gateway City and click "Finish Adventure." Then, on the home page, find the pet who has the eggs and click their inventory. On the next page you'll be able to select the eggs (or items) and click "Send to Main Inventory."

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

@Cyrinus

Finishing a pet's adventure doesn't erase their progress, it just puts them on standby :)