Where the quantum mechanics enters in BEC, why can't (non-interacting) classical particles condense?
"Classical particle", or atoms do condensate. At high temperature $T$, the atoms are far away in the so called gas phase. When the temperature decrease, they will undergo a phase transition and condense to liquid. At even lower temperature, it becomes solid and the atoms are closer together. All of these three phase has clear phase transition temperature. As the temperature get even lower, the atoms are typically even closer. But "classical particle" can't occupy the same spatial location.
However, there are particular type of particle called boson (say photon, or quasiparticle, not necessary atom) that have the special properties that they can occupy same quantum states. So at low enough temperature, all of the particles can occupy the same ground state, and hence the Bose-Einstein Condensate. It is a very clear phase transition because typically most of the particle are not in the ground state unless the transition occurs. For example, dilute atomic bose gas has BEC transition around 500nK, while some quasi-particle has such transition at room temperature.
I think that classical particles won't occupy the same same states, as if they are rigid like billiard balls, they can't occupy the same position. But if you drop this constraint and put them in a potential well, they probably would do something similar to a BEC (all occupy the lowest energy level as you drop temperature).
However, the difference in a BEC your states are also quantized, so the behavior probably won't be quite the same, in particular the heat capacity will probably be different.
they can, it's called a regular old liquid.
a BEC is when individual atoms lose their identity, in a sense, and the wave functions overlap to display group behaviour.
bad for BEC if the particles condense classically and form a liquid = pool of something other than a BEC, prolly just a liquid pool of classical particles.
at least as I understand it.