Is an RF connector required for an NFC Tag?
The chip's data sheet says this: -

And, realistically, 13.56 MHz is not particularly problematic with most reasonable types of connection.

The antenna ports appear to be balanced but I don't know for sure.

It's primarily a magnetic field communications method so impedance balancing isn't needed. It would be if it were a dipole antenna (for instance) but, it isn't. It's a near-field magnetic field antenna.

Twisted pair vs not?

I would twist the wires between the PCB and the antenna just to cut down extraneous magnetic fields that would otherwise be lost if twisting is not used.

Would I need a balun or can I connect it straight?

No, a balun isn't required because it's a near-field magnetic field antenna. Balun's are needed when connecting (for example) a balanced dipole to a coaxial feed. They are not needed here; it will work equally effectively with a balanced drive or an unbalanced drive. Not sure what the chip provides but, it's largely irrelevant.