McGrath --- followed your link to 'Green Grow The Lilacs'... Irish soldiers singing about the Red, White & Blue? I don't think so.... plus this, from the same link:

Oh yes, the story about this being the basis for the Mexican epithet of "gringo" is most likely pure fakelore. RG

If I were of a marching disposition I think I could do it to 'Green Grow The Rushes-O' (but fortunately I'm Not!)

BJ