Can the runner in baseball advance after he is hit by a thrown ball and it goes out of bounds?

One answer:No - the runners may not advance beyond where they were - or more specifically, where they were about to go - at the time.

A batted ball that hits a runner is a "dead ball."

Another answer:If, however, the batted ball first touched a defensive player or umpire prior to being touched by the runner, the runner is not out and the ball is live.

If the umpire calls interference on the runner, the ball is dead, the batter is awarded first base, and all runners return to the base they last occupied at the beginning of the play (not to the base they were about to go to at the time).