Some games (ROMs) will automatically save your progress when you exit the game correctly. To exit the game correctly, press START+SELECT together to return to the RetroPie UI.
Certain games won't save your game automatically when you exit the game. You can manually save your game by pressing SELECT+R1 together. On the GPi, R1 is the shoulder button on the back of the unit.
The Retroflag GPi Case looks very similar to the original Game Boy, but it definitely doesn't play like one. The case features a 320px x 240px full-color display and requires a Raspberry Pi Zero.