PLEASE help me understand the 144 Visa Free Travel and how do I explain it to United Airlines?

Look, the greatest mistake is to try to get an answer on the details such as visa or visa free transit from the customer support of some airline. Simply speaking they will answer that you have to ask relevant Chinese institution or that you do not have right to do that what you have planned to do. Like this they get rid of their responsibility and in the same time they transfer all the burden of obtaining the information on their passengers.

OK, I can see that United is flying from US with a direct flight to Beijing from San Francisco. So, if your flight is:

San Francisco - Beijing - Hong Kong

And if you do not have in this exact segment, because other flight details are not important, any other landing of any kind, you are definitely and undoubtedly eligible for 144 hour visa free transit valid for a wide Beijing area.

You really do not need to explain anything to to United your right, but when you arrive at check-in in SFO you will just declare that you will travel to Hong Kong via Beijing using 144 hour visa free transit. The personnel employed at the check-in counters for a flights that go directly to China are usually very well versed about this scheme, but if you see that a person that is doing your check-in is not exactly the best in this, you will just tell her/him to check at the Timatic. Just be relaxed because you will be allowed to board the flight. There is no doubt about it. All you need now is to do not think about these details any more and to appear at the airport on the day you have your flight. Have a nice journey.