I'm a little confused about why these exist and what their distinction between regular positive voltage regulators are.
It seems to me like if I wanted to get -5V from -12V and 0V, that I would connect a normal 7V regulator, with -12V to its GND and 0V to its \$V_{in}\$.
Is the current flowing in the other direction? The positive regulator would fry or cause an open circuit?
Answer
Simple answer: You would get 5V in your case, but referenced to -12V instead of ground. In other words, you would have -7V, not -5V. In addition, the regulator would only source current onto the -7V rail, not sink it as would be expected for a negative voltage.
If you want to run some circuitry from 5V between -7V and -12V (the -12V will be the ground for this circuitry), then you can use a positive regulator as you described. If however you want to run some circuitry between ground or higher and -5V, then you need the negative regulator.
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