Screw & Glue Subfloor and Screw Underlayment
Yes, do both.  To avoid squeaks, use screws (not nails) everywhere for the flooring.

I had a very squeaky kitchen and entry-way floor, which I wanted to tile over.  I kept the existing plywood subfloor but screwed it down to the joists using drywall screws.  I used a LOT of them, one every 4 inches or so along every floor joist, and it eliminated all of the squeaks.  

Then we put down Hardibacker-type cement board and screwed it to the subfloor.  We didn't worry about screwing the cement board to the joists; just screwed it to the subfloor.

That was two years ago.  No tile has cracked or shifted, and no squeaks have returned.  Definitely use screws and not nails.