Skullbriar's last ability only works if it has that ability in the zone it's moving from. For example, with Yixlid Jailer ("Cards in graveyards lose all abilities") on the battlefield, a Skullbriar with a counter on it in a graveyard loses that counter when it's put onto the battlefield. Conversely, that Skullbriar moving from the graveyard to the battlefield would retain that counter if Humility ("All creatures lose all abilities and are 1/1") were on the battlefield; if Skullbriar then left the battlefield with Humility still on the battlefield, it would lose the counter.