Here's what I've found out from National Health Insurance Administration Ministry of Health and Welfare. http://www.nhi.gov.tw/English/index.aspx
I gave them a phone call at 02-2706-5866 extension 2221 for help in English. I spoke to two people Mrs. Zhuang and Iris, both of them were really nice and helpful.
It seems that foreign parents, with ARC or APRC, who have children born in Taiwan must wait a 6 month period until their newborn children are eligible to get medical coverage for their kids. The 6 month wait starts from the date your child gets their ARC.
The only work around is if the newborn children are employed. If they are able to get jobs i.e modelling then you might be able to get them National Health Insurance based on the fact that they are paying taxes and earning an income.
This seems like a law that really needs changing.