Can an HDMI external recorder such as the Atomos Ninja Asssassin record 4K from the FS700R?
Or can the FS700R only output 4K via SDI?

Sorry I didn't see this thread earlier. No, the FS700 cannot output 4K over HDMI at all. It can only output 4K over SDI.

Its important to note that the FS700r does not output actual "video" over its BNC port. Its actually only 4k raw sensor data protocol. Its not an SDI video standard. Its just a Sony propriatary data stream.