How to detect when the lid of a box is loosened (as soon as the screws are unscrewed)?

You seem to be looking for a switch that activates "when someone starts to unscrew the screws"

This can be accomplished by positioning the tip of the lever of a microswitch at the end of the screw's shaft. As the screw is turned it travels up its threaded shaft, and the tip gradually releases pressure off the microswitch, which eventually tips the switch.

It requires some fine grinding and drilling, of course, to get the switch in the right position. But it does provide the exact action you want. You can trip all 4 screws with this: when any screw is opened, its switch is activated.

As an aside, this is a common anti-tampering trick in exterior alarm sirens. There is a double housing: an inner lid and an outer shell. When one (or two) of the outer shell's screws are unscrewed, their microswitch is activated, thus opening the normally-closed 24hr anti-tamper circuit and triggering the sabotage alarm.

The conventional approach to the simple switch to detect box lid removal is to use a microswitch. These come with through mounting holes so that you can use a couple of machine screws to mount the switch to the inside edge of the box. They also come with a convenient lever that the cover can press down when the cover is closed. Finally many microswitches have push on lugs that allow easy wiring with crimp quick connect terminals.

Here is a picture of the typical styles. As you can see one has just the sensitive button.

^{Picture Source}