I don't know if this is the Paddle Your Own Canoe you are thinking of or if it has great historical significance, but this one is in L.E. Wilder's By the Shores of Silver Lake. (Little House of the Prairie is only one of her interesting books.) This is a song she learned from her Pa and sung on the American frontier. I think it was sung to the tune of Rosin the Bow and may indeed have Irish origins.

1.I've traveled about a bit in my time
And troubles I've had me a few
But found it better in the ocean of life
To paddle my own canoe.

2.My wants are few, I care not at all,
If my debts are paid when I die
I drive away strife in the ocean of life
While I paddle my own canoe.

3. The love your neighbor as yourself
As the world you go traveling through
And never sit down with a tear or a frown
But paddle your own canoe.

I hope the lines are broken. I just now trying.