I think you can get this permit online, or at the Chunghwa travel agency, which is like an unofficial Taiwan government office. I've just seen a relative do it online, it seemed easy enough. But you have to be born in China (mainland, HK, Macau) to do it online.
The HK re-entry permit is used to enter HK itself, I don't think you can really use it go to anywhere. And you don't need one if you have another document that can let you enter HK, such an an HKSAR passport or HKID card.
It's pretty much only of use for mainland immigrants to HK that haven't got right of abode yet, and perhaps lost their ID cards, or those that are under 11 that don't have an ID nor passport.