When stars and whatnot are pulled in through an event horizon could it come out through a white hole?

Unlike black holes, white holes have never been observed. The effects of black holes on surrounding matter has been observed many times. Wormholes and white holes seem to be permissible by Einstein's Field Equations, but that doesn't mean they are a physical fact. As Donut Tim suggests, there are at least two theories regarding the center of a black hole. Probably the best way to think about the singularity that lies there is to think of it as an area of space in which all the laws of physics as we know them break down. Just as we cannot describe the exact moment of the Big Bang, so we cannot describe what happens at the singularity. This leads many physicists to suspect that there is an intimate relationship between the two objects, though we will probably never prove it experimentally.