I could be wrong but for the team star you have to complete the stage without using any units from the last time you won, as can be seen on the mission screen.
Ex if a mission had baba and bird, you can't use either of them and have to figure out a way to still win, either on the normal version or the extra mission, both work for this star.
Some mission are really tricky and you need to use enemies pathing to clear them, ex without bird or me to destroy blocks, if an enemy can do so you need to position them with other mechs to indirectly destroy blocks.