I am SIngaporean and I would like to apply for Z-visa. What documents do I need in order to get it?

I think it's hard for you to apply for a China work visa in China. You'd better apply for the work visa from the China Visa Application Service Center in Singapore.
The documents you have to prepared include your valid passport with at least 6 months valisity and 2 blan pages, a completely filled visa application form, Employment Permit for Foreigners issued by the Ministry of Human Resources and Social Security, "Invitation Letter of Duly Authorized Unit" or "Confirmation Letter of Invitation" issued by an authorized unit in China and Employment Permit for Foreigners issued by the Ministry of Human Resources and Social Security if you will stay in China for more than 6 months.

Could I apply it in Hong Kong and do the medical checkup in China instead?

I think it's also possible for you to apply for a China work visa in Hk, but not sure whether the officers of the visa office of PRC in Hk accept your medical checkup in mainland China. You can try to contact them to confirm this through tel. 852-21066303