I lived in SZ when I was pregnant. I looked into having my baby in HK, but it was too much stress with having to scramble to get papers (passport, visa) after the baby is born and then having to stay in a serviced apartment with a newborn by myself while my husband worked in sz.
You need the birth certificate to get the passport and then that to get the visa. As soon as you arrive back in China, within how long I can`t remember bc we `violated` it, you need to register your baby at the police station where you`re registered.
I ended up having him in GZ at the VIP Birthing Centre. Had a good doctor, awesome room, overall good experience and for a fraction of the cost of HK. PM me if you want and details.