how to remove these screws without taking a huge chunk of wall with them
I believe you are describing an "expansion wall anchor". Here is what the manufacturer states: 


  Primarily used in drywall, the hollow wall anchor is a medium duty fastener. Once the body of the anchor is installed, the screw can be removed and items can be changed without disrupting the anchor. The tip has a drive point to pierce drywall easily.


Here is a picture.

The only way to remove any expanding or toggle-type anchor (with minimal damage) from the wall is to push the anchor into the wall hollow and spackle the opening.