Newell, in "Games and Songs of American Children," originally printed in 1883, has a section on Jenny, which seemingly started out in Scotland as Jenny (my) Jo (sweetheart). Since the description is earlier that the 1927 book, which apparently borrows from Newell, I reproduce newell's text:
In one version, here is his description of the game in which Jennia dies.
A mother is seated. Miss Jennia Jones stands behind her chair, or reclines on her lap as if lying sick. A dancer advances from the ring.

"I've come to see Miss Jennia Jones,
Miss Jennia Jones, Miss Jennia Jones,
And how is she today?"

"She's upstairs washing,
Washing, washing-
She's upstairs washing,
You cannot see her today."

The questions are repeated to the same air for every day of the week, and the reply is that Miss Jennia Jones is ironing, baking or scrubbing. She is then represented as sick, or worse, and finally is dead, which announcement is received with sighs of deep grief. The dancers of the ring then discuss the costume in which she will be buried:

"What shall we dress her in,
Dress her in, dress her in,
What shall we dress her in-
Shall it be blue?"

"Blue is for sailors
So that will never do."

"What shall we dress her in,
Shall it be red?"

"Red is for firemenn
So that will never do."

Question repeated for other colors-
"Pink is for babies,
So that will never do."

"Green is forsaken,
So that will never do."

"Black is for mourners,
So that will never do."

"White is for dead people,
So that will just do."

"Where will we bury her?
Under the apple tree."

After the ceremony of burial has been completed, the ghost of Miss Jennia Jones suddenly arises-

"I dreamt I saw a ghost last night,
Ghost last night, ghost last night,
I dreamt I saw a ghost last night,
Under the apple-tree."

The ring breaks up and flies with shrieks, and the one caught is to represent Miss Jennia Jones.

W. W. Newell, 1883 (1965, Dover), "Games and Songs of American Children, No. 11, pp. 63-66. Newell provides music for two tunes, one from Maryland, No. 174, pp. 243-244.

Newell remarks that the game was once a love play, in which the heroine pines because her cruel parents refuse an offer for her hand.

I think the Dover paperback is still available. An excellent old book.