How can I fix a squeakly tile bathroom floor?

Sound produced from flooring comes from movement between pieces of subfloor or between the subfloor and joists. The answer is to find what's moving and stop it. Typical tile floors have 3/4 inch plywood attached to the joists with a 1/4 inch smooth surface like sheet luan over that.

If you are able to temporarily remove the tile, screw (not nail) the subfloor down at 4 inch intervals in both directions and replace the tile.

If the top layer cannot be temporarily removed, you will have to wait until the top layer can be replaced. If the luan top layer is replaced also, it's worth the effort to apply beads of construction glue between the two layers before screwing the floor down to the 3/4 inch.

The above is assuming you have some kind of flexible vinyl tile. If the tile was a rigid tile like ceramic, slate etc., your problem would be cracked tiles or cracked grout. An under-floor has to be totally stable like a concrete slab to accept that kind of flooring.