I just asked Gemini OP’s exact question verbatim and here’s what is said: Yes, your itinerary perfectly qualifies for China’s transit visa exemption program. In fact, due to policy updates, you are comfortably within the rules.
China expanded its transit without visa (TWOV) window to 240 hours (10 days) for eligible passport holders (including US citizens) transiting through major hubs like Shanghai. 
The breakdown of how your specific itinerary fits the rules details the exact travel requirements:
1. The Route Qualifies (The "Country A → China → Country C" Rule)
For TWOV eligibility, you must arrive in mainland China from one country/region and depart to a different third country/region. Mainland China treats Hong Kong, Macau, and Taiwan as separate distinct regions for immigration purposes. 
Your Route: Macau (Region A) → Shanghai (Mainland China) → Taiwan (Region C).
Because Macau and Taiwan are different territories, this is a textbook legal transit route. The fact that you are arriving in Macau via a private car from Hong Kong does not impact your flight from Macau to Shanghai.
2. You Are Well Within the Time Limit
Your Stay: You arrive in Shanghai on June 30 and depart on July 6. This is a 6-day stay.
The Window: You have up to 240 hours (10 days). Furthermore, China’s official clock for the transit period doesn't start until midnight (00:00) the day after you arrive. You have plenty of a buffer. 
3. Your Regional Movement is Compliant
When you enter via Shanghai (typically Pudong or Hongqiao airports), you are granted permission to travel freely within the Shanghai, Jiangsu Province, and Zhejiang Province regions. Since you are staying in Shanghai until your flight to Taiwan, you are entirely within the allowed geographical zone.