It is an interesting question. So much so, it also got me wondering. In the search, I came across a small museum. The Ralph Foster Museum College of the Ozarks in Point Lookout Missouri has in its collection four firearms believed to be from the show. They also have the truck.
I have been in touch with the curator and the shotgun they have is marked "The Continental" He says it has exposed hammers, 24" damascus barrels, overall length of gun is 40 1/2" with s/n 26xxx, Patented June 20, 1883. He says of this shotgun that it appears to be in very good condition and
"We feel it is the gun used in the show because it along with three other guns from the Beverly Hillbillies were donated to us by a friend of Paul Henning."
When checking out "The Continental" it appears to be a Belgian made gun. Some pics
here
I'm sure there must have been more important things to do lately, but you gotta love "Granny". Anybody been to the museum?
Cheers,
Jack