No, IanC is right about Pills. In fact two songs set to the tune appeared both in the 1706 edition and the edition of 1719-20. The tune itself was in a number of publications of the period, including Playford (II, 2nd edn., 1714), Walsh (1719, II, 74) and so on.

Both Pills texts are in the DT (see links above): also the 'origins' thread which contains more detail than this old thread recently revived.

'The Beggar's Opera' used the tune for the song 'Were I laid on Greenland's Coast'; the 'Recruiting Officer' song doesn't appear.

See Simpson, The British Broadside Ballad and Its Music, 561-3, and Bronson, Traditional Tunes of the Child Ballads, I, 14-16, for more on the history of the tune.