There is no way that the Mainland-Taiwan relationship will get better any time soon. It's only going to get worse. I would not count on CKG working.

Here's what you should do to minimize your losses:

1. Call Eva and see if you can change the flight to HKG-TPE-USA (and I'd guess there's a small but non-zero chance they will waive the fee if you explain the immigration issues)
2. Book a separate, fully refundable direct HKG-USA flight
3. Have the PRC citizen fly to SZX, and then take the land border into HK. Mainland passport holders can transit in HKG for 7 days when leaving the country, and exiting the mainland at the Hong Kong land border tends to have the fewest issues due to the higher volume and familiarity with Hong Kong transit rules. However, when exiting the mainland, show mainland departure immigration the direct HKG-USA flight. Departure immigration may not allow a HKG-TPE-USA routing.
4. Once in Hong Kong, cancel the refundable HKG-USA direct flight. Check in for HKG-TPE-USA at Hong Kong airport normally. Hong Kong immigration will not care about taking the HKG-TPE-USA flight.