It should be noted that squadrons cannot directly own stations. Squadrons support factions, and it's the factions that control the stations. Factions are run by NPCs, whether it's a "player-made faction" (PMF) or a procedurally-generated faction. Players do not have direct control over a faction's behaviour, only indirect influence.
The Squadron members are presumably allied to the faction they are supporting, so they will probably get better-paying missions once their faction is in control. There are also things you can do to help keep that station under that faction's control (like trading and selling exploration data), which you can't do until they actually control the station. Finally, the faction's government type might make certain things legal that were previously illegal, and/or vice-versa - which will presumably be of benefit to the players and squadrons pushing that faction.