Colonel Young offers to leave the damaged shuttle with the people who choose to stay behind on the condition that the members of the military return to the ship.
He also states that the shuttle cannot break atmosphere.
If the shuttle can't break atmosphere, it can't be flown back to the Destiny and therefore it would have to be left behind whether or not Young's condition is met.
Rush calls the star system "the Solar system". That would be Earth's star system, the one with the Sun in it, not some alien star system billions of light years away in some different galaxy. Those are called "star systems".