Here is the full list of all 13 HellStar Squadron achievements.
Beat the game with Emerald.
Beat the game on Hard.
Attain a x100 combo.
Unlock Niji.
Beat the game with Azure.
Destroy a total of 500 enemy ships.
Clear a stage without getting hit.
Attain a x50 combo.
Destroy a total of 250 enemy ships.
Beat the game with Citrine.
Beat the Game with Niji.
Beat the game on Veteran.
Beat the Boss Rush mode.