My father, too, would sing just two lines of this, and I've been trying to figure out it's origin. He was an Army man, so maybe this was some sort of crass joke men in the military would pass around? He was in the Army in the 50s.

The way I heard it was:

"Oh there are no Chinese laundries where the River Shannon flows...
The Irish are a dirty bunch, they never wash their clothes..."

if there was more, I don't remember. This was not derogatory in our home, btw... my dad's grandparents came from Ireland.