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