I will be mounting a 4”x4” electrical box at the back of an under the sink cabinet. A wire for a garbage disposal will be fed through a hole directly in the back of the box. Another wire will be fed through the side hole from the adjacent dishwasher, and I intend to put a connector there. Do wires feeding from the wall directly into the cabinet and through the back of the box require a cable connector? I will also be wiring another box for 240v wall oven using #8 cable the same way - thinking same rule would apply. Would it?
Cables require a cable clamp - the clamp can be external to the box (which requires installing the clamp and cable before mounting the box in the case described, unless you have access to the back of the box when mounted) or internal to the box (where it adds to the count for box fill), but you can't just stuff the cable into the box through a bare hole and call it good.
There are two functions of a cable clamp - one is strain relief - if the cable is pulled on, the clamp takes that strain, and the electrical connections of the conductors do not. The other is preventing damage to the insulation from contact with the sharp edges of the hole in the box.
Wires in conduit do not require clamps. Cables require clamps.