that the sub flooring below the tile was only nailed down and the sub floor is dancing up and down on that nail causing the sqeak and being flexible causing the in flexible grout to break. What I would do is very easy and will not require a major do.
Get a wood chisel and remove the grout around ground zero, then carefully remove that one tile. You may have to break only one to get it out so that you can remove all the others in whole pieces. If so, get a hammer and with eye protection, wallop that tile dead center and remove the pieces. then remove others in that area until the sub floor is bare and clean. Make sure you go back far enough.
Now get some 2 inch galvanized or stainless screws and apply one screw every four square inches. Make sure that the screw heads are burrowed or sunk below the surface a bit.
Now, lay back the tile with thinset or tile mastic. Then wait for 24 hours, then grout. this should do it.