Can you provide a source for this accusation? I'm Chinese and I have never heard of this before. But China is a very, very large place with a very diverse group of people and specific practices may be isolated to certain regions, provinces or even villages. But to make the sweeping generalization that Chinese people believe this... is blatantly false. Even if you were to say that the majority of Chinese people believe this, that would still be horrendously inaccurate.

Three things to keep in mind:

Firstly, sometimes when you hear of practices that seem unusual to western sensibilities, they may only reflect superstitions held by a very small number of Chinese in a particular area during a particular time period. Even if what you hear was a superstition that existed in China, many of these superstitions died out during the Cultural Revolution.

Secondly, don't believe everything that you hear. I'm not accusing you of making this up, but there are people who will make up bizarre practices without any solid historical evidence in order to slander Chinese people. Additionally, these type of rumors are also made by asians about other asians in order to portray them as barbaric. For example, I have heard of Koreans accusing Japanese of eating human baby fetuses, and Japanese claiming that Chinese do, and nearly every other combination of nationalities making these claims. If you hear about some strange claim that "Chinese people do", it may not be false but take it with a grain of salt until you have some proof. And if your proof is that your friend told you, then you want to be a bit suspicious.

Thirdly, just like in America and the west, there are cults and religious sects in America that have strange practices. Many of these popped up in the 19th century and more recently with the Chinese government's relaxation on religion. These practices don't reflect the beliefs of vast majority of Chinese people in China or abroad. Sometimes you will find a Qi Gong guru or a Feng Shui master who makes a specific claim as part of his/her teachings. But besides that person and his/her students, the belief is not held outside of that group.

Sorry if this seems like a very strong reaction to your question, but I have heard way too many accusations that "Chinese people believe ______" (insert any bizarre practice that satisfies your need to believe that Asians are strange and uncivilized). If you can provide some proof of your claim, I'd be very interested to see it.

edit: I did some digging around on the internet trying to find out if anybody else made the same claim as you did. There is a post on /r/AskTrollX about this, but it seems like a troll post (which your post may be as well.) I found this page, which is a "chinese medicine" organization from Florida. They don't claim that Chinese people don't shower during menstruation. In fact they claim that American women didn't in the past: "American women are not taught proper diet or exercise. Our grandmothers and great grandmothers understood the importance of taking care of themselves during their menses. Modern women ignore these strictures as old fashioned or sexist. During the menses, a woman should not drink cold drinks, swim or bathe (showers are okay)." Take from that what you will. Finally, I found a Yahoo answers where longliveabcdefg makes the claim that this was done a long time ago. To me, this is akin to asking why Europeans only take 2 baths their entire life - applying a practice from the middle ages to today.