I'm guessing the OP wishes to shower before exiting the airport?
The EVA lounges are in
departure area of T2, MH arrives in T1 (correct me if I'm wrong, somebody...). I don't think they will allow arriving PAX to use them, even if arriving with EVA (unless transiting).
More have lounges in both terminals... here is the link to the T1 lounge.
http://www.moregroup.com.tw/t1/e-index.htm
It doesn't say it has a shower, but I suppose it does. If all else fails, and he can get to departures (is that possible?) he could pay into this one.