Changing the "black sheep" rhyme is over reach.
A better comparison with the Giant would be "eanie meanie minie moe, catch a tiger by the toe. If he hollars let him go."
Originally the "tiger" was "n****r".
Racism, and Jim Crow, are a recent painful memory. So its necessary to change the rhyme.
But the ethnic wars between Scots and English are in the distant past.
Though I do remember, as a child, being confounded as to the WHAT Jack's nationality had to do with anything? Its not made clear that he is English the beginning of the story, and its not obvious what his being English has to do with anything in the story. In fact it wasnt until reading this thread that I knew "English as opposed to WHAT? American? French?Chinese?. Okay ..so now I finnally know that its "English" as opposed to "Scottish".

That figures. A lot of Elizabethan era politics was secretly lampooned in rhyme, and is still preserved like bugs in amber, in modern nursery rhymes. Humpty Dumpty was a big oversized cannon commissioned by the king of England that was mounted on a castle wall, and it blew up, and "all of the kings horses, and all of the kings men couldnt" put it back together again.
I remember the eenie meanie mo that was definitely racist.