Hi John,
The tune my mother used was similar to 'Jimmy Crack Corn and I don't care', just the first 2 lines repeated, with perhaps going up on 'chi' and just coming back down the scale on the 4th line of your rhyme.

Just a thought, the 'J' on jingo is very close to the 'ch' sound so if that was a misheard 'chingo' it wouldn't be that far removed from many versions above.