JFK-PEK-MFM-HKG-JFK

American airline staff is obviously too ignorant to look at Timatic that is the only source of reliable information regarding the visa requirements and similar. Their database is just a portion of what Timatic could say in the most authoritative way.

By the way, your itinerary is (was) definitely eligible for 144 hour visa free transit in Beijing.

What to tell you? In case that this was your definite itinerary and there were no other stops involved between New York and Beijing and especially between Beijing and Macau, you should definitely start a procedure for your reimbursement that could potentially lead to a lawsuit because you have really good proofs that you were right.

Thank you!
Our flight is for Friday morning.
We were getting nowhere on the phone so my husband actually went to JFK airport and had to inform the agent about searching TIMATIC database AND putting Macau as our final destination, not Beijing.
Hence the title TRANSIT without visa. After about an hour the agent was finally able to find all the information.

As has been mentioned countless times on this site, it is always a complete waste of time calling the airline about this.