Horace will give you a Quest for you to save him. You will meet him in Pirate Voyage, locked in some kind of trap. If you free the gremlin that is in a chamber next to him, he will tell you that the place Greene is locked on really is a trap he made for Hooks's mechanical pirates, the he will open the door to free him. To complete Horace's Quest, you have to talk to Greene when the door is open. If you don't save Rigger Greene in the Pirate Voyage boat ride, he'll become a Lost Character inside the Blot.
In the second game, he appears to have grown a white beard like Beluga Billy and now has a grumpy personality. Since his cousin Moody lost his memory, he gives the player the Thanks for the Memories quest. To complete this quest Greene's path, you have to take pictures of certain place where Moody and Greene can go for adventure, or wait to simply choose one place for them to stay. The places you can choose from are: Rainbow Falls, Disney Gulch, Fort Wasteland, Floatyard, and Autotopia. Each of these places will get you a reward with Rigger Greene involving Collectible and Generic Pins. If Moody gets his memories from Ostown back in his side of the quest, then Rigger Greene will have to go back to Disney Gulch by himself.