Since you claim to have a "Z" visa, I will assume you are not holding a Chinese passport.
As a U.S. passport holder, you will be allowed to enter Taiwan without a visa for up to thirty days (no extensions permitted) as long as you have a passport valid for at least six months beyond the date of entry into Taiwan and have a confirmed return or onward air ticket. You must have already met any additional visa requirements for the next destination, if applicable. If your passport is valid for less than six months from the date of entry into Taiwan, you may apply for a landing visa that has a duration of stay of no more than thirty days. The cost of a landing visa, including processing fee, is NT $5140 (currently equivalent to approximately US $165.00). Extensions are not allowed. You also have the option of applying for and receiving a Taiwan visa prior to arrival in Taiwan. The cost including processing fee is US $164.00.
Please Note: If you enter on the visa waiver program (30 day stay, no extensions), you will not be allowed to extend your stay or change your visa status.
If you arrive in Taiwan with a valid passport and valid Taiwan visa, you may be admitted for up to 90 days even if your passport is valid for less than six months. If you apply for a landing visa upon arrival, you will be admitted for 30 days or up to the date your passport expires, whichever comes first. Visit the Taipei Economic and Cultural Representative Office (TECRO) ’s website for the most current visa information.
If you do not have a US passport, then please check the following website for your requirements and the address of Taiwan Embassy in Hong Kong :

http://taiwan.visahq.com/embassy/Hong-Kong/