You can make such a pushbutton using a spring mechanism and electrically conductive foam. The more you compress the foam, the more current it conducts (the lower the resistance). With the right kind of foam, it could also act as the spring mechanism. The push button would press on one metal contact, and another contact would be on the opposite side of the foam.
A circuit to drive your LED would be quite simple You could do a decent job with the foam, a couple of resistors and a transistor and a power source, such as a battery. Multiple LED's would be only a little more complex. Essentially, it would be a bar graph driver circuit. You can do four LED's with a quad comparator or up to ten LED's with with one chip, such as a National LM3914.
Conductive foam isn't very predictable, because the resistance depends on the thickness and area of the foam, as well as the characteristcs of the type of foam you find. You'll need to know enough electronics to understand how to bring such a ciruit into the desired range of operation and to calibrate it.