Good question, Morwen, because the "folksong" might actually be an adaptation of the Belafonte record.

I googled "what makes your head so red", which produces 38.800 hits - wow! Most of them refer to two sources from US-American folklore, also to be found at Mudcat. The first exists in many variants about various species of birds, e.g.

Woodpecker, woodpecker,
What makes your head so red?
Been peckin' at the wood most all of my life
It's a wonder I ain't dead, I ain't dead, I ain't dead.

and the other is

Tenbrooks said to Molly what makes your head so red?
Runnin' in the hot sun puts fever in my head
Fever in my head O Lord fever in my head

The first one compares the colour of the bird's feathers to a human face red with exhaustion (i.e. filled with blood, as when blushing), the second one compares the colour of a horse's fur either with a sunburn or with genuine fever, both visible in the face of a human.

In my opinion, this proves to me that my first guess is well supported by US language, apart from being the only one that makes sense to me for the Burgess/Belafonte song. Diane Browne's interpretation is about hair. The meaning of the Jamaican version is yet to be deciphered, it may follow either of these interpretations, or a third one, or it may even be gibberish due to lack of understanding US language.