Drill through the screw anchor using the drill fitted with the 1/4-inch bit. Make sure to drill into it several times, so that you puncture the end inside the wall in several places.
Drill into the lip of the drywall anchor at the "3 o'clock" position, meaning at the right-most side of the anchor. Penetrate the rim as many times as you need to to loosen that side of the anchor.
Drill into the lip of the drywall anchor at the "9 o'clock" position, meaning at the left-most side of the anchor. Penetrate the rim as many times as you need to to loosen that side of the anchor.
Insert the needle-nose pliers into the screw anchor, grabbing the inner sides.
Pull, making sure to pull backwards and not back and forth. You will find that the anchor comes out either in one mangled piece or in several sections.