I know I can't supply 12v from the board but can I use an H-Bridge I have to control a stepper motor (don't know the number off hand since I'm not at home at the moment).
Why would you do that? Just use a relay. Use the arduino to trigger the relay to pick, and have the 12V wired to the common and the other connetion to the horn.
Test with a simple button or something else, if you get a signal the horn should activate.