Can a minor who is US citizen travel back to the United States from Jamaica with a parent on a expired passport ?
I cannot answer definitively, and possibly there is some wiggle room depending on the agent. But with today's immigration environment, you are taking a risk of significant inconvenience if the child does not have a current passport. I would not take that risk.
I suggest a visit to the passport section of the US Embassy, or a US consulate. They will be able to answer the question, and probably to renew the passport. I would not expect rapid service for a new passport, though if it is an emergency, they may be able to expedite.
A US citizen cannot be denied entry to the US no matter what; as long as they make it to a US port of entry and can satisfactorily prove their US citizenship (and an expired passport is pretty good proof), they will be let in.
The problem is how to get to the US. I am assuming they are flying. US does not (yet) have pre-clearance facilities in Jamaica, so the airline must check that they have required documents before letting them board, and US citizen children arriving by air are required to have a valid US passport.