You asked "And, can someone explain the process of the Liver Flush? Where and why does apple juice and apple cider (either one) come into play?"
Apple juice, or apple cider, contains maltic acid which will soften the Gallstones present in your gallbladder and liver. Softened Gallstones will pass more peacefully through the gall duct.
Drinking the grapefruit juice-olive oil mixture "activates" a lot of bile flow. Of course, the magnesium sulfate ( Epsom Salts ) will cause the gallbladder and gall duct to relax so big stones can move out more easily.