I'm not sure how all this works but currently, I live and work in China and would like to visit Zurich. Is it at all possible to apply for a Schengen visa outside my home country and if so where can I go? In China or Hong Kong?
If you're a legal resident of China (ie with a Z/X etc visa and a resident permit) you can apply for a Schengen visa at one of the consulates nearest to you. Depending on which country you will apply at, and where you live, you might have to apply to a consulate in a specific city, as consulates usually have a region of competence.
However, unless you live in Hong Kong (or, often, Macau), you cannot apply there, as you're not a resident there.
Hong Kong almost defiantly not.. ( but see the explanation for china )
In China, if you have sufficient reason you theoretically can - and I am writing this because a business colleague of mine just did that same thing a few weeks ago ( in GZ ).
The Official policy says "NO", an applicant should apply in their home countries - but for every rule there are exceptions and there are some cases where the consulate can make an exception. These reasons are not exactly specified but can be anything that requires urgency for example and are left for the consul judgment.
In the case of my colleague, the Visa-Center has referred him ( with appointment ) directly to the consulate , where he had an interview and was asked for the reason why he can not return to his home country to apply, what is he doing in China, why he needs a visa etc. etc.
After the Interview the Consul signed a form for the Visa-Center and the application was received and visa granted ..
Just like @user67108 wrote in the related answer, if you have the legal permit to live in china ( example work permit / temporary residency ) - then you should be able to apply there..