Here is the full list of all 13 HellStar Squadron achievements.
Beat the game with Azure.
Clear a stage without getting hit.
Destroy a total of 500 enemy ships.
Attain a x100 combo.
Destroy a total of 250 enemy ships.
Attain a x50 combo.
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.