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