My mother and I will visiting Shanghai on 144 hour visa but flying into/returning to Los Angeles.

You cannot fly USA-China-USA without a visa. If you attempt to do so, you will be denied boarding at your origin. To qualify for 144 hours transit without visa, your route must be (for example) USA-Shanghai-Hong Kong-USA or USA-Hong Kong-Shanghai-USA. Any third country would work instead of Hong Kong, there aren't "approved destinations".

Thank you so much for this information. The qualification was confusing but we understand and adding Hong Kong.

So our trip as indicated below would follow the same idea as what you're explaining here and we would be able to qualify for the 144 hr Visa Free Transit, correct?

So here's our flight plan -
Chicago >Shanghai (Stay for 5 days)
Shanghai> Bangkok (Stay for 10 days)
Bangkok > Shanghai (4 hour layover) > Chicago

Yes, it is correct and you qualify for the 144 hours.