Yes, it was on quite a few occasions that Johnny Cash met up with Richard Nixon. The session captured on this tape recording is from July of 1972. The meeting was picked up the the hidden microphone system planted in the White House Oval Office and several other places around the Executive Building. In fact, the same taping system, as you might suspect, that later led Mr. Nixon to leaving office.
The Super8mm footage of Johnny Cash playing comes from the Richard Nixon Library. The footage was all taken by members of the White House staff and later impounded as evidence during Watergate. You can see much more of the footage and the story behind it in the excellent film, Our Nixon.