You don't get negative stats, but some interesting things can happen.
The developers don't intend for a Pokemon to have negative stats, so the data associated with stats are stored and read as unsigned integers by the game (that means: whole numbers which can only be positive). Even if you play around with the numbers in the game's memory, it simply won't read the number as a signed int (that means: possibly negative) because it's programmed that way.
What happens if you try anyway? The short answer is integer overflow (or perhaps "underflow" if you go below 0), which sounds interesting but generally isn't a problem with good foresight and data validation. Unfortunately for the developers, they let the Pomeg glitch slip for several generations. This is a technique where you use EV-reducing berries to make the game calculate your Pokemon's current HP as <0 or "negative". Since the data is unsigned, when the game tries to read the HP stat later, it will interpret it as being extremely high (near or equal to the highest number that can be stored in the memory space).
Here's a demo of some things you can do with the Pomeg glitch in Emerald. (I was caught off-guard by the song choice and now you will suffer through it with me.)