Korea to Shanghai to Xiamen to Los Angeles

We will be traveling this coming June with US passports. We will be leaving Jeju, South Korea and headed to the United States. I have found a flight from Jeju to Shanghai on Spring airlines, and then another flight on Hainan airlines from Shanghai to Los Angeles with a layover in Xi'an. Is it possible to book these flights using the 72 hour or 144 hour travel exemption? We will have to collect bags in Shanghai and recheck in for the Hainan flight, but would not leave the airport before the connecting flight. If possible, we would prefer to leave the airport in Xiamen as we would have a 24 hour layover there and would love the opportunity to see the Terracotta Warriors and museums. Please advise if we would be eligible for the travel without visa with this itinerary.