Best way to remove cement backer board?
 

Screw heads are stuffed with caulking and cannot unscrew. What is the best approach to take down backer boards?

crow bar/wonder bar, elbow grease!!!

No matter what kind of caulk it is you should be able to drive a bit into the screw head. Or try to remove the caulk with a razor knife. It will save alot of time.

I'm confused by the issue. Why was caulking put on this? Are you sure it's not thinset?
As previously stated, a metal bit should be able to be inserted into phillips head screw.
Why do you need to remove the board?