Although the recent U.S. shutdown did not last long enough to have a negative effect on the Asian market, an extended shutdown would have a negative impact on it. U.S. debt comprises an extensive amount of Asian funds. If the U.S. government was shutdown for an extended amount of time, and no government funds were moving, then it would inevitably have a negative impact on the Asian market.
Yes, the shutdown of the US government would have affected the Asian market economically, had it lasted for much longer than it did. Many Asian nations, most notably China and Japan, have invested in the US economy through bonds, and had the US defaulted on those investments, it would have shook the economic world of Asia.