This song has been in the Cub Scout Song Book for as long as I can remember - which, as somebody pointed out to me in another thread, is a long, long time. The song would fit almost perfectly into the genre of "camp song" except for one thing - it's just too nice a song. It must have been made up by an adult leader of a youth group. Judging from the content and structure of the song, I would venture to guess that it was made up by a harried Den Mother who didn't know what to do with the rowdy Cub Scouts who were chasing around her basement one wintry afternoon. So, she took a traditional tune, "Turkey in the Straw" and added hokey words with a strained rhyme (hence, "Continental Soldier," Tim). And, to keep Timmy from hitting Johnny, she added ridiculous hand motions to the song to keep Timmy's hands busy. And, when it found that there was nothing whatsoever naughty or discriminatory about the song, the Boy Scouts of America of New Brunswick, New Jersey, put its imprimatur on the song and printed it in the Cub Scout Songbook and later in the Boy Scout Songbook.
It became a true camp song after that. That same bratty, obnoxious Timmy who occasioned the song in the first place, took that innocuous little ditty and added the words that qualified the song to be printed in "The Erotic Muse."
Now, all this is musings from an aging former Scout leader, and may or may not be the truth. However, I have seen many other really dumb songs formed in the same way - including the later transformation into naughtiness.
The song appears in the 1969 revision of the Cub Scout Songbook and the 1970 revision of the boy Scout Songbook, but not in the 1963 Boy Scout Songbook. I'd like to know if it appears in print any earlier.
It also appears in the 1988 edition of the "Prairie Home Companion folk song book," with a second verse:

Do your ears hang low?
do they wobble to and fro?
Can you tie 'em in a knot?
Can you tie 'em in a bow?
Can you throw 'em over your shoulder?
Like a Continental soldier?
Do your ears hang low?

Do your ears hang high?
do they wave up in the sky?
Do they crinkle when they're wet?
Do they straighten when they're dry?
Can you wave 'em at your neighbor?
With a minimum of labor?
Do your ears hang high?

-Joe Offer, Retired Cubmaster-