If I purchase a refurbished iPhone can I still get Apple Care?

The only OEM factory refurbished iPhones available are those sold directly by Apple in their online refurbished and clearance store. Anything claimed to be “refurbished” by any other seller (bestbuy, Amazon, a carrier, etc) is at best refurbished using salvaged/used parts by a third party refurbisher, or is nothing more than a used device.

No, those phones are not refurbished by Apple, you will have to go with Best Buy's extended warranty.