Travel between Shanghai and Shenzhen

No, unfortunately this is not possible. A travel under 144 hour visa free transit implies only one stop in China or better to say, you are not allowed to have two mainland Chinese stops in the row.

You can somewhat adapt your itinerary to look like this, for example

Europe - Shanghai - Taiwan - Shenzhen - Hong Kong (or Europe)

If you adjust to this itinerary you will be eligible for 144 hour visa free transit twice, in Shanghai and in Guangdong transit areas. When I say Guangdong I have in mind Guangdong province where several large cities are located like Shenzhen and Guangzhou among others.

Just to conclude, it is essential to have in mind that the immediate points before and after your mainland Chinese destination have to be in two different countries or eligible territories. Eligible territories are Hong Kong, Taiwan and Macau.

Thanks. Is there a minimum amount of time I need to be in Taiwan or Hong Kong before going back into China to Shenzhen?

No, there is not such a requirements. You can be in Taiwan for literally an hour. It is just important that your flight after Shanghai touches down in Taiwan and then proceed toward Shenzhen.

If you want to proceed from Shenzhen or to be precise Guangdong province you can proceed from there to Hong Kong if you want based on 144 hour visa free transit by a direct high speed train to West Kowloon station in Hong Kong. If you decide for this option, you should just have in mind that you will have to buy your train ticket online in advance, print it out and have it at hand ready to be presented to relevant authorities. Also, you will have to take exactly that train you have bought your ticket for.