By progressing daily through the game, you will eventually be able to get every Moshling, without secret codes, however you will only be able to hatch them after a certain amount of Moshlings have already been hatched. Consider these codes as quick access, rather than exclusive access.
To enter these codes just go onto the Moshi Egg Hunt app and go to the code system which is located on the top right of your screen. From there just enter the code of shapes to unlock that Moshling instantly.
More codes will be added shortly.