Not really "songs" perhaps but there are a fair number of old time fiddle dance tunes with ducks or geese in the titles that used to have some words associated with them.   Best known may be "Ducks on the Mill Pond" and the "Old Gray Goose".

Fiddlers Companion suggests words to the Ducks on the Mill Pond were:

    "Ducks in the millpond, Geese in the ocean;
      Hug them pretty girls, If I take a notion.

      ***

      Cho: Lord, Lord, gonna get on a rinktum,
          Lord, Lord, gonna get on a rinktum.

      ***

      Ducks in the millpond, Geese in the clover,
      Jumped in the bed, And the bed turned over.

    ***

      Ducks in the millpond, Geese in the clover,
      Fell in the millpond, Wet all over. (Lomax)

      ***

      Rain come and wet me, sun come and dry me,
      Step back pretty girl, don't you come nigh me. (Tommy Jarrell)"

All of which sound like pretty standard dance tune floaters.

Same source also has words for one version of Old Gray Goose being:

    "Johnny Gordon lost his cow,
      And where do you reckon he found her?
      He found her up that rocky branch,
      With a hundred buzzards around her.

      ***

      Cho.:
       Look here, look there,
       Look away over yander;
       Don't you see that old grey goose,
       A-smiling at that gander.

      ***

      Johnny Gordon lost his wife,
      And where do you reckon he found her?
      He found her up that rocky branch,
      With a hundred men around her."

However I can't really make those words (particularly the chorus) fit with the Old Gray Goose fiddle tune I know so it may be a different version.   The verses and chorus don't really seem to match that well so again maybe these are floating verses