I know that there's no evidence at this point for "white holes" however would it even be mathematically possible for a black hole to be connected to a white hole (total opposite so everything would be expelled, after some really extreme physical conditions)? Maybe with a wormhole connecting them? If this was even possible (if the black hole or connection could actually be created and be stable enough), would that matter be expelled into a different universe, etc? Maybe even a different region of spacetime? Just curious, as it would be a cool idea.
Dear Kahtrijn, white holes are microscopically the same objects as black holes, and it's guaranteed - by the second law of thermodynamics - that all macroscopic processes occur in the way as they do in black holes and not white holes (the latter are time-reversed of the former).
However, if you don't care about the second law of thermodynamics that prevents black hole size from shrinking by emitting large objects (which objects?), then you may write down any configuration in general relativity you want. In particular, it's easy to connect white holes to black holes. After all, an "eternal black hole" is a solution that is doing nothing else.
The connection between a white hole and a black hole goes in a different direction than you seem to expect, however. It's a connection between the past and the future. It's because the (neutral) white hole singularity is a spacelike singularity in the past, and the (neutral) black hole singularity is a spacelike singularity in the future. To connect those singularities, you need timelike (rather than spacelike) trajectories.
One doesn't need any wormhole to write down an eternal black hole. It may sound cool to combine so many different holes but it is totally unnecessary: the wormholes are independent objects from the black holes and white holes.
Lubos Motl's answer is correct. However, the so-called maximally extended Schwarzschild solution to the equations that govern gravity describe what may also be considered an untraversable wormhole. It connects two "universes" (i.e. asymptotically flat regions). But the connection does not last long enough for a signal to pass through it.
For a suitable observer this solution essentially describes an object that is a white hole in the past, a short-lived untraversable wormhole in the present, and a black hole in the future.
[Note that astrophysical black holes (as opposed to the theoretically permissible black hole solution considered above) result from stellar collapse and do not involve wormholes or white holes.]