Squeaky stairs, fix from below?
I repaired my poorly-build stairs in just this manner recently. Here's what I did after removing the drywall:


Run a bead of high quality construction adhesive along all joints and laps. Every point of contact between stringers, risers, and treads should have one or two beads, depending on access. Use a protected finger or tool to press the bead into the corners and gaps, spreading it and enhancing contact. 
If the stairs are at all springy, add 2x4 or 2x6 lumber to the stringers. You could even add two members to the center stringer. Use construction adhesive and 2-1/2" gold construction screws. 


The result is a stone-solid staircase with no noise whatsoever. If you have any noises remaining after allowing a night for the glue to set up, tackle them on an individual basis. Chances are more glue and maybe some shims will do the trick.

Reinstalling drywall on the underside is trivial for your hangers. I doubt that they'd even upcharge for it, especially if they bid to tape that area anyway.