When I have trouble cutting a thick-skinned squash, I stick the point of a French chef's knife in as deep as I can, then use the back of the blade as a lever to push the edge around as far as I can.
In theory I should probably try to whack them with a thin-bladed cleaver, but I'm a sissy and don't want to miss and take a finger off or gouge my kitchen counter or knock the cutting board off the counter.
I'm practicing with a cleaver outdoors, on things brought home from the fruit stand dumpster. 7/8th inch plywood, cleaver, whack-whack-whack, feed the compost heap and then count my fingers.
