'The Juice of Barley' appeared in Playford's Dancing Master both under that name and as 'Cold and Raw' (to different dance figures). 'Cold and Raw' was a "Scotch song" (a generic term that refers to style not origin) written by Thomas D'Urfey; its popularity led to the name of the tune to which it was set (originally 'Stingo, Or the Oyle of Barly', in Dancing Master 1651-1686, but altered in 1652 by John Hilton to the form used for 'Cold and Raw') changing.

For Playford examples, see  http://www.izaak.unh.edu/nhltmd/indexes/dancingmaster/

Claude M Simpson, The British Broadside Ballad and Its Music, goes into detail in the section 'Stingo, or Oil of Barley, or Cold and Raw', listing various songs that used forms of the tune. Transcriptions of his musical examples can be seen at the late Bruce Olson's website in abc format:  http://www.csufresno.edu/folklore/Olson/

Although Chappell printed the words of 'A cup of old Stingo' set to the 'Stingo' tune (Popular Music of the Olden Time, I, 308-9), that was supposition on his part, as the song originally appeared without tune direction. "Stingo" was a common term for strong ale. If there was a song from which 'Stingo' got its name, it seems not to have survived; though rather a lot of songs used the tune in one form or another, and under a series of names.