Welcome to SciFi Fight Club!
Our initial Fight is a match between two assimilating species to see who the victor will be. Cast your vote and list your arguments to participate in the sport of SciFi fandom.
If I recall correctly the Borg used the nanite injection tubes on various shipboard system consoles as well as humans. I don't recall any dialog supporting this but I have memory of a borg plugged into a console in a few scenes.
The Thing organism was active at the cellular level. Assimilating cells by engulfing and replicating a cell.
The Borg nanites imply they are cell assemblers which operate at the molecule or atomic level. Borg nanites could then disassemble the Thing cells molecularly. Breaking the Thing cells down into their base atomic structures.
The only way a Thing could grab the upper hand is if there was a shortage of nanites and the thing cells dominated the playing feild. But, even if the nanites were temporarily overrun since they are inorganic and the thing cannot replicate inorganic components the nanites would still reverse the process or disolve the cells.
Thus, The Borg would win.