As Raymond Elliott has just found, you're not allowed to drown them. Nor are you permitted to blow them up with explosives or electrocute them, suffocate them or feed them into a garden shredder. And you certainly can't launch them from a clay trap and shoot them in mid-air.
The officially sanctioned DEFRA method for despatching a cage-caught grey squirrel is to release the creature into a stout Jute sack; it will immediately thrust its head into one of the two bottom corners. Take a tight grip on the sack and administer a sharp blow to the head with a Hickory hatchet handle or similar implement.
However, DEFRA officials may now be prosecuted by the RSPCA for such advice; the squirrel-huggers are advising that the only acceptable way to kill a captive grey is to chauffeur it to an official Squirrel Euthanasia Centre and 'put it to sleep' to the sound of soothing Mantovani lift-music after a last meal of hazelnuts.
The problem with the RSPCA's method is that you can't eat 'em after. I've grown quite fond of fresh grey country squirrel prepared in the Ghanaian fashion, with chilli, onion and a chopped hard-boiled egg. A better 'elevenses' has yet to be found.