mary,
Not that IMDb is the absolute source, but it is fairly good, and the only joint venture they show with both Johnny Cash and John Wayne is a TV variety show special from 1970 called Swing Out, Sweet Land. Aside from the one user comment, there is no other detail, but from what I can discern from that one commentary, the two Johns did not even appear together.
Hopefully, someone else around here might be able to get you closer to your answer.
Chester 
I like some of the Clint Eastwood movies - The Outlaw and Josie Wales is my favorite. I enjoy quite a variety of Roy Rogers movies as well, and his TV series. As a kid, I used to enjoy the Lone Ranger, and the Cisco Kid along with his pal Pancho. And for some beautiful horse footage, there's nothing like The Man from Snowy River.
For some reason, I've never particularly enjoyed Gary Cooper or Jimmy Stewart in cowboy movies, although I've liked them in any number of other movies.
Chester
