Here is the full list of all 15 Galaxy Squad achievements.
Complete any quest in the game
Complete game in campaign mode on any difficulty
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
Train a character to level 7
Defeat warlord boss
Defeat a swarmmother boss
Defeat a cyborg brain boss
Complete story campaign with the redemption arc
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