What interests did Austria-Hungary have in China at the time?
Very interesting question! This is one I had to do a bit of digging on. The very few scholarly articles I've found seem to agree on a number of major points that would have contributed to the Austro-Hungarian intervention in China:
The presence of Austro-Hungarian missionaries and diplomats. One newspaper in Slovenia (then a part of A-H) wrote in its editorial page that "today we must face the cruel fact that the Chinese are single-handedly overruling the international law with their bloody acts carried out on innocent civilians, who are not responsible for the events caused by the global politics." The intervention was seen by many members of the public, as well as the Catholic Church, as necessary in order to protect the lives of Europeans then living in China.
Strategic interests: Chinese coaling stations, especially those in Tianjin, were extremely important for the massive battleship fleets then in vogue among the European Great Powers.
Matters of prestige: Some of the articles I consulted agree that Austria-Hungary got involved despite its lack of economic and political interests in China simply in order to be a team player. Sending troops to China as part of a Euro-American-Japanese mission would have been seen as a matter of Great Power prestige. Participating in the intervention signaled a state's international reach and interests, which at the turn-of-the-century would have been of chief importance to certain A-H politicians and diplomats. In the past few decades, Austria-Hungary had been kicked out of Italy and pushed out of Germany. It's only imperial "acquisition" was Bosnia, occupied in 1878. Thus, participating in the Boxer suppression would've been seen as a way to increase A-H's international profile.
International gamesmanship: Finally, I read one article abstract which claims that A-H's intervention in China was part of a larger policy that aimed to divert Russia's attention eastward, instead of westward, where Austro-Hungarian and Russian interests were increasingly coming into conflict. The author doesn't clearly demonstrate the link between Austria's small contribution to the intervention forces and how this was supposed to "distract" Russia away from the Balkans. I was only able to access an abstract of the article, so perhaps the author goes into further detail in the body of the text.
I hope this helps! It seems you've struck on a very interesting and not very-well explored corner of Austro-Hungarian history. I find that the extant scholarship adequately explains the short-term influences on the decision (protecting European lives, strategic access, gaining economic concessions, etc) but what I was able to find doesn't really get at your larger question of why intervene at all. I'm hoping that in the future we might get an historian who's interested in this larger structural, cultural question. If you've got German-language skills and an interest in doing history, maybe this could be your research question!
EDIT: forgot to link my sources