Liver flushes are recommended pre and or post fasting. The gallstone issue is not because of low calories or fast weight loss. It is that when one fasts since one is not consuming fats bile production has all but stopped and is at a minimum creating conditions favorable to the formation of stones.While doing liver flushes is always a good idea it is not a crucial issue on a 10 day MC as that it is the "minumum" lenght for an MC.
It is NEVER a good idea to do a liver flush in the middle of a fast. Just plain common sense. Dumping a bunch of oil into a digestive system at rest is just not a good idea. 1st it puts a screeching halt to the fasting process and 2nd all that oil dumped into a system at rest has a great potential of making you quite ill.
Ideally one should wait till after one breaks the fast and you are back to normal eating and having consumed some fats and having normal eliminations. Again just plain ole common sense to allow the system to normalize rest and rebuild after any cleansing procedure. Also this waiting for the system to normalize allows for the build up of bile to have a more productive flush.