Einstein's equations do not predict blackholes. Blackholes are simply compatable with his equations.
This does not mean that blackholes may be incompatable with other physical laws, notably those of quantum mechanics/field theory and those of thermodynamics, which is why it is theoretically interesting to try to derive the quantum and thermo properties of blackholes to find either a contradiction or an interesting property which one might try to observe from earth.
Someone who says they do not believe in black holes either
1) does not believe Einstein's equations, of which they are solutions.
2) believes that other physical laws prevent the occurrence of these solutions.
The first paper on this Bose-Einstein condensate stuff poses another solution of the GR equations in which the point singularity is replaced with a different structure, the BEC. The math seemed all on the up and up.
(BTW the Schwartzchild solution doesn't really have a singularity. The singularity is an artifact of the coordinate system used, just like the singularity of latitude and longitude of the earth -- and we do believe in the north and south poles here, right? Kruskal exhibited coordinate systems in which there is no singularity.)
So what we have is a new analytic solution to the GR equations (and there are not many, so this will undoubtedly make it into graduate texts in the next decade).
The bad news is that the geometry around a gravastar is identicle to that around a blackhole. It is just different when close to the phenomenon, so all that business about terrible cosmic death at the hands of a gravitational giant is still there.