With optimal conditions can we supercool any compound or is it that just water and a certain set of others can be supercooled? If not, why?
Supercooling is a general phenomenon and it is not limited to any particular substance, although the degree you can supercool something will vary. The idea in thermodynamics is that at the melting point the free energy of the solid and the liquid is equal and a lower temperature will favour the formation of the solid.
However, when there is only liquid present, you will first have to form a small crystal nucleus which is not as stable as the bulk solid i.e. has a higher free energy. This is basically a kinetic barrier which prevents forming the solid.
The formation energy of such a nucleus will depend on the interfacial energy of the crystal-liquid system. In a substance where this interfacial free energy is very little there won't be too much supercooling.
So called 'drop tubes' have been used for decades to study the nucleation rate vs temperature of supercooled molten droplets of (mainly) metals in vacuum or inert environments. An (admittedly old) example is 'A review of long drop tubes as a supplement/alternative to space experiments', R. J. Bayuzick et al., Advances in Space Research 4(5) 85-90 (1984). Nucleation studies of supercooled semiconductors (Si and Ge) was also studied back in the 1980's. Googlingh 'drop tube recalescence' will find more. Overall, I'd say the literature is quite full of supercooling on any number of different materials.
It is interesting to note that supercooling of a liquid is much much easier than superheating of a solid - the problem is decidedly not symmetric in the general case. The difference between nucleating a critical volume of solid in a liquid droplet is not the same as initiating melt at the surface of a solid, where there are unsatisfied bonds because of the interface.