To buy a squad base, you need 30000 reputation and 100k caps for the small base, 200k for the big one. To join an existing squad, you need to be invited by someone from that squad. The invite is sent by an NPC inside the squad base, and to view / accept / decline your invites you can talk to the building manager. In the lawyers base, the building manager is the guy in power combat armor in an office to the right of the main entrance. In the family base, the building manager is a black guy in metal armor on the second floor in an office to the north.
We are planning some changes to the squads and squad bases in the upcoming days. The system works already, but it doesn't offer much except, well, a base for you and your friends.
You can only be in one squad/base at a time, so to join a different squad you need to resign from the one you're currently in.
There are apparently some issues with the squad bases, I saw some error logs in the server window. It's being looked into.