From what I understand of the link: Segura made multiple minor league baserunning mistakes in this play. Segura was leading off second base and could have attempted to steal third base regardless if the pitcher threw the ball home or not, he just made his first baserunning mistake and went too early for third and saw that the pitcher could have easily thrown him out at third, so then Segura ran back to second before the pitcher could attempt to pick him off at second. Braun made it to second base, and in that situation Braun is automatically called out because two baserunners cannot occupy the same base at the same time.
Braun should not be called out. Once you advance, the base is yours. The runner who was previously on the base (Segura) must advance to the next base before being tagged out. You are right about two runners not being allowed on the same base, however it is the lead runner who is to be called out when it happens. This is baseball 101, the sort of call little league umps like myself see all the time. I could give the major league umps some slack since pro players don't make such a rookie mistake hardly ever, but it still doesn't excuse the fact this is very basic rules knowledge in the game.
Incidently, once Braun takes second, Segura can be forced out by tagging the ball at third - no need to chase him down.
I'm amazed the major league umps missed this.