The saying is more-or-less proverbial. Clifton didn't necessarily coin the phrase, but he did write the song: he always had an ear for a good catchline (another of his songs was called 'Where There's a Will There's a Way') and a good tune, many of which he picked up and used with great success. One of the most successful songwriters and performers of his day in both Britain and America, it's hardly surprising that many of his songs became part of popular currency.

This thread, revived after a sleep of some years, is one of the least informative ones we have here on this particular song; it's usually that sort that people who wander in via a search engine pick. Of the list of links above, neither the DT entry (the verses from Laura Ingalls Wilder, which somebody always quotes when the topic comes up) nor the ancient (1998) discussion Paddle me [sic] own canoe credit Clifton; but see also thread Harry Clifton Songwriter (2002) which also quotes a text and contains a lot of useful information.

American broadside and songsheet examples have already been quoted; see also  Bodleian Library Broadside Ballads for examples printed in England, Ireland and the USA, with a couple of parodies and a couple written for the same tune:

Paddle Your Own Canoe