Would Visa be required for 4 hour layover, and then a 24hour layover in shangahai?

If your entire itinerary is exactly like this:

US - Shanghai - Taiwan

Taiwan - Shanghai US

And you don't have any additional landings, especially within mainland China, your first stop in Shanghai will be perfectly covered by 24 hour visa free transit and your second stop by 24 or 144 hour visa free transit. I understand that you have written that you will have 24 hours layover in Shanghai during your second stop, but in case if you have a minute less than 24 hours, you will be eligible for 24 hour visa free transit like as you were during your first stop and if you have a minute longer than 24 hours, you will be eligible for 144 hour visa free transit. In any case, you don't need any kind of visa. You are advised to print out your flight tickets and to have those printouts with you. Also, you should book your accommodation in Shanghai in advance and to have a confirmation of your booking in the form of printout.

Basically, this is it.