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