Poet, singer, cartoonist for Playboy, composer, writer and generally extraordinary Shel Silverstein wrote "The Unicorn" for his book "Where the Sidewalk Ends" and the first recording of the song appears on the 1962 issue "Inside Folk Songs"

Yes they are from Ireland but The Irish Rovers (Will Millar, Joe Millar, Jimmy Ferguson and George Millar) formed in Toronto, Canada. The group charted with the cover in1968 when they peaked at seven on the pop chart and number two adult contemporary. According to Wikipedia that's Glen Campbell on guitar.