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