The Mode S 24 bit address is quite distinct from the squawk code (and the related flight ID). The 24 bit address is "just a number", and the only official significance is that every aircraft has a unique number. If you're into computers and know anything about local area networks, then the Mode S address is the equivalent of the low level hardware address of your Ethernet card - it is used at the lowest level to make sure the packets are received by the correct destination, but has no significance for higher level protocols. In the radar environment, the 24 bit address is used purely by the interrogating radar to uniquely "paint" a single aircraft. What is then received by the interrogator (and displayed to the controller, etc) is the usual squawk code, altitude and so on. So, from an ATC operational point of view, it's still the squawk they track, not the 24 bit ID.
Thus, a Mode S transponder user inputs a squawk code in much the same way as a Mode A/C transponder user.
In answer to Alex's comment, there are still only 4096 codes for the squawk. The difference is that, because the computer that is running the radar can reliably track each individual target (because it can discriminate using the 24 bit ID) there is no problem with squawk overlaps - you could cope with two targets flying near each other, both with the same squawk, and the computer would never get confused about which was which.
A more recent innovation though is that in addition to the classic squawk code, new Mode S transponders can also transmit Flight ID. On an airliner that would be the flight code, like EZY1234. On a GA type that is usually the registration, like G-ABCD. This makes life a lot simpler for the ground radar system. For backwards compatibility though, the squawk is still always required.