"Quantum-tunnelling composite" is a rubber material impregnated with metallic particles. Its resistance varies as a function of mechanical pressure.
Basically, put a piece of QTC membrane between a pair of metal plates and squeeze it, the resistance of the QTC will decrease with an increase in pressure. This allows more or less current to flow from one contact to the other.
This stuff works remarkably well, I've used it as the lower half of a resistor voltage divider and you can get very smooth and reproducible results over a wide range of pressure. You can even use it to form a crude but effective speed controller for (very small) motors.
In your case you'd use the ADC of your microcontroller to measure the voltage and convert that into an integer value that quantifies the pressure on the button.
It's widely available and not very expensive if you only need a couple of square inches of it.