For me, I notice I have to play as each squad member I have created for him to earn xp/rank up.
They don't rank up on their own when they are AI. I am only playing squads in the private online mode with my friends though. But I don't think they ranked up when I have been played against in offline mode.
You will earn Squad Points in Squad Mode. The points will be under your account and can be used by any squad members you selected. The points can also be used in Squads or in Multplayer. For the Squad Members, you will need to rank them up 1 at a time. Once each soldier reach Max Level 60, they will prestige and will no longer receive XP. This means you will need to use the next soldier to rank up and prestige again in the game.