I think this song is trad but probably not originally Irish (might as well get in ahead of the others). There's a song "John John the Grey Goose is gone and the foxy's away to his den-o" which I think came up on a thread last week that is at least a first cousin of this one.

There's also a macaronic (half Gaelic,half English) version of the song that I learned in school, but then I went to school after the Clancy's were popular, so this doesn't rule out the posibility that they composed it.

The tune I think is definitely Trad and shows up in a number of versions of the famous descriptive piping piece "The Fox Hunt" which has been played as a set piece by uileann pipers for at least 100 years if not longer.

The correct spelling is Madirin Rua (with a "fada" on the last I) which means litterally the little red dog, but is the gaelic name for a fox.