I'm a little confused by your question. Do you mean "Han China" as in the Han Dynasty? Or do you mean Han, as in the majority ethnic-linguistic group thatm akes up modern China?

Either way, I can confidently say that by varying degrees and definitions, China did adopt Buddhism as a state religion at different points in history, and to the differing whims of this Emperor and that bureaucracy. That said, it would sort of be a misnomer to say "China adopted Buddhism as a state religion," but more accurately, "China adopted various aspects of Buddhism as parts of their imperial cult." For example, the concept of Chakravartin, important for Indian concepts of universal Kingship (the Buddha was to the religious realm as the Chakravartin is to the worldly realm), was imported and super-imposed onto the Emperor.

Moreover, the places you mention either adopted Buddhism and Buddhist concepts of statehood as a result of Chinese influence (Korea, Japan, Vietnam) or long afterward (Tibet, Mongolia).