Flooring – Should I replace nails in a plywood subfloor with screws to the joists

Use screws instead of nails to hold the plywood down. If you can, screw it into the joists, not just the existing floor. You can also put down some glue betwen the plywood & subfloor to prevent the plywood from moving at all. Use tongue-and-groove plywood if possible.

If that doesn't help (and you should be able to do that & check to see if you still have squeaks before putting down the hardwood) you might need to beef up the joists, or add additional bracing, from underneath. Not so hard if the ceiling below is open, a lot of work if it's finished.

You should shim under the loose joists with steel shims, not cement, because cement causes wood to deteriorate. Large galvanized washers and small steel plates can be used as shims. The framing block in between the joists are called bridging and in an old house I would space them about 4' on center. You can off set each row by 1 1/2" which will enable you to blind nail through the joists which minimizes toe nailing. Use 3/4 tongue. And groove ply glued and screwed to the joists and bridging. I would use pl400 for glue and 2" exterior screws spaced about 8" apart. If a second layer of ply is desired you should also glue and screw it the same as the first sheet. Try to hit the joists. Use a chalk line to mark the joists. If a solid wood finished floor is being installed 1 layer of 3/4 plywood is all you need.if a ceramic tile,stone or wood laminate floor is going in use 1/2 ac plywood over the sub-floor.