Here is the full list of all 15 Galaxy Squad achievements.
Complete game in campaign mode on any difficulty
Complete story campaign with the redemption arc
Train a character to level 7
Complete any quest in the game
Complete a game in free mode on any difficulty
Complete a game on hardcore difficulty in any mode
Survide 20 turns in survival mode
Survive 50 turns in survival mode
Defeat warlord boss
Defeat a swarmmother boss
Defeat a cyborg brain boss
Complete story campaign with the servant arc
Complete story campaign with the traitor arc
Complete story campaign with the deal arc
Complete story campaign with all 4 arcs