How can I remove just one tongue-and-groove subfloor board?
Are the tongues necessary? Can I just take my circular saw right
  between the boards, cut them way, and then screw them back down
  afterward?


You've been misinformed.  The point of tongue-in-groove planking is to keep the floor boards from twisting, slipping and sliding against each other and squeaking (or squeaking more in your case).  You still need to secure a tongue-in-groove floor to the subfloor or joists.

Joists almost never squeak on their own.  Squeaking comes from two pieces of wood rubbing against each other - usually because a floor hasn't been adequately secured.  

Now if you DO need to cut away the flooring, then you should cut along the seams, and then buy Hardwood Floor Spline and rout out the old tongue so you end up with two grooves facing each other, and use the spline to rejoin the flooring when you reattach it.

But BEFORE you do that, I'd get a bunch of 2" coated screws and drop them through the subfloor into the joists below and see if that fixes the squeak.  It's subfloor so you don't care about the face, and if it doesn't work, they're easily removed.

*Tip - use a hammerdrill when putting in the screws