black hole - Eternally collapsing objects?

With respect to "eternally collapsing", they are probably referrring to the fact that in the reference frame of an external observer, gravitational time dilation prohibits matter from ever reaching the event horizon (the "surface") of the black hole.

Denoting the radius of the event horizon $r_mathrm{S}$ (for "Schwarzschild radius"), time runs slower by a factor of $(1 - r_mathrm{S}/r)^{-1/2}$ for an observer at a distance of $r$. As $r$ approaches $r_mathrm{S}$, this factor goes to infinity, i.e. the observer will never reach $r_mathrm{S}$

However, this is only for the external observer; the falling observer would cross the horizon and reach the center in a finite time.

With respect to "not being stable", you're probably right that they are referring to Hawking radiation which presumably makes the black hole slowly "evaporate". Near $r_mathrm{S}$, pairs of virtual particles are being created and can be turned into real particles by the gravitational field. If they avoid falling into the black hole, energy is taken away from the hole, reducing its mass.