To be honest, originally I did not want to answer this sort of question. It is to be noted that "stupidone" himself referred to the assumptions in his question as "absurd".
Now for the answer:
A rigid rod:
1) The material objects of our world are made up of atoms, which have mass and interact by electromagnetic forces. Pressing the rod would lead to a displacement of some atoms closest to the pressing finger, this displacement occurring at a speed lower than the speed of light. The displacement leads to a rearrangement of electromagnetic fields in the rod, this change propagating at the speed of light, this change leading to a change of forces on further atoms, causing their acceleration and respective displacement, and so on. As a result, a mechanical wave is travelling along the rod. The atoms therein are moving slower than the speed of light. Therefore, the end of the rod next to the button will move after the light signal has reached its corresponding button.
2) Suppose the rod made of a more exotic material, like nuclear matter (leaving aside the issue of stability of the rod shape here). Then instead of atoms and electromagnetic forces, you will have protons and neutrons (which have mass), and nuclear forces plus electromagnetic forces (which are carried by fields disturbances of which propagate at the speed of light). So, eventually, it is a situation as under 1.
3) Let's be even more exotic and disregard the details of the composition of the rod. The rod is rigid. But what does that mean then? It, by definition, means that as soon as you press one end, the other end moves in the same way as the pressed end does. But in this case the definition necessary to understand your setup (i.e., what does "rigid" mean) already contains the answer to your question.
4) What can a rod like in 3 be? It either is something utterly different from the physical world we know, not even composed of constituents, maybe. Or it is made up of particles which interact by forces we do not know but which propagate faster than the speed of light. Such particles themselves then are also not known to us.
Your question therefore boils down to: If I assume things significantly different from known physics, do I predict results significantly different from the predictions of known physics?