Most Chinese do support their government, publicly. This final word is the key here. The Chinese Communist Party, or the CCP, does not like anyone bad mouthing them. The more negatively someone talks about the CCP in China, the more likely it is that they are being watched by the police. A lot of people within China are not happy with their government.
This is easily understandable when you think about the image of China. Most likely, you immediately thought of flashy lights, lots of skyscrapers, modern buildings, etc. That’s what you’d get in Beijing, Shanghai, and other large cities along the coast. However, further inland, many people are still in poverty, and there is no way for them to get out of poverty because the local governments are just as corrupt as the national government.