It depends on a number of factors. Obviously they'd have jurisdiction over something physically present on their territory. However, they would also have jurisdiction over things done abroad with criminal effects in their country, like DDoS attacks, or that affected their citizens, like a Chinese manufacturer shipping defective, toxic products that harm their citizens. International law has a jurisdictional principle called the protective principle that allows a sovereign to treat attacks on its citizens as attacks on the sovereign, and confers jurisdiction over the malefactors.
However, from the wording, you can tell these dipshits know they don't have even a tortured argument for jurisdiction of any sort over an American under these circumstances. The whole pathetic rant is a threat of "take that down you bad man or we'll put you on the list of bad men!"
www.youtube.com
