*It sounds as though you have a decent subfloor. What I would look at
is the joists. How long of a span do they have without support?

Something else to consider is that the previous owner may have put down
a fast and cheap tile job to sell the house. The tiles may be popping up
because they were not installed properly.

I think you are right about the tile job. No screws were added to the
existing subfloor to prepare it for tile. The ply wood is too thin and
doesn't have nearly enough screws. Now, the Hardiebacker website says
that 5/8" ply is the minimum to put the cement board on top of. Does
that mean if there is nothing else underneath it? So if I remove the
1/2" layer then screw the 3/4" layer to the joists every few inches so
that it is very solid, could I use 1/2" ply under the cement board? I
would still have over 1" of wood underneath the cement board.

I really appreciate everyone's replies.
Thanks
Dante