Certain eggs will unlock GBA mini-games when hatched. You'll have to download it into your GBA.
Game Egg Location
Chu Chu Rocket Pirate Island 2
Nights Circus Park 2
Billy Hatcher Hyper Shoot, Easy Dino Mountain 2
Billy Hatcher Hyper Shoot, Hard Sand Ruin 2
Puyo Pop Blizzard Castle 3
There are a few ways to get more emblems, including beating all missions with an "S" rank, collecting every coin and unlocking the whole Egg Gallery.
Play as Bantam Scrambled:
Beat Blizzard Castle Mission 4.
Play as Chick Poacher:
Beat Dino Mountain Mission 4.
Play as Rolly Roll:
Beat Pirate Island Mission 4.