The Dunn Hawken is not a "Hawken barn gun". It is a heavily used and "frontier" repaired J&S Hawken rifle similar to other known Hawken rifles.
It's presently in the Jim Gordon private museum in Glorieta, NM, and is pictured in his three volume book set,
Great Gunmkaers for the Early West.
James Dunn J&S Hawken
BallBoy linked a thread that provides about all that is known on the history of the rifle and its previous owners, so I won't repeat it here.
The Dunn Hawken in its original configuration probably looked very similar to the Ed Louer Hawken rifle pictured in Baird's first book in Plate 3 and Plates 36-40. This rifle is also in Jim Gordon's collection.
Ed Louer "J&S Hawken" - barrel markings illegible
As
hawkeye2 pointed out, the Dunn Hawken is missing its butt plate and has some very crude replacement of wood for the original butt plate return and the toe. In addition to this work, the original breech has been cut off, re-breeched, and barrel set back. In the process, the nipple has been installed directly into the barrel rather than in a breech bolster or drum and nipple. The rifle was originally a full stock and rather crudely cut back to half stock.
The patch box is probably not original to the gun, but was added at a later time, possibly when some of the other crude repairs were performed on the rifle.
We don't know much detail on the rifles history. Jim Gordon writes that James Dunn was a wagon master on the Oregon Trail. Another author referred to him as mountain man. I've not been able to confirm either of these claims with primary documents and think he was just a pioneer. It's not known if the damage and repairs were made on the trail or during its use in Oregon.