Can my common-law partner accompany me in China on a S1/S2 visa?

He can join you in China, but it is impossible for him to get a S1 since you are not married. For S2, it is hard to tell and you need to consult your local Chinese embassy/consulate for consulting.

For most cases, the application with your same or similar situations can only apply for a Tourist with single or multiple entries to stay in China. This type is much easier than getting a S2.