Paris - Shenzhen (two legs flight, 3h stopover in Beijing) - Taipei

No, you are not eligible for 144 hour visa free transit in China because you have two consecutive stops within mainland China in two different transit areas. However, you can be eligible for 24 hour visa free transit if:

1. Your first port of entry to China is Beijing and Shenzhen comes at the later stage
2. Your entire transit time from your scheduled landing in Beijing until your scheduled departure from Shenzhen is shorter than 24 hours.

In any other case, you will have to obtain a Chinese visa in advance being it a transit or a tourist one.