I have auburn hair and the same stripe, although I color it. I'll tell you the best advice I got from a hairdresser, though. She said not to use red hair dye, because if your hair is already red, it will turn purple/burgundy/magenta. She advised me to use light golden brown, and I tell you, it works! I color it myself, and it turns out almost exactly what it was before the gray started. The light golden brown brings out the red, so I end up with auburn. You might need a slightly different shade if your hair is flaming red, perhaps a lighter blonde, but nonetheless you can avoid that burgundy hair.

For the record, though, I think the skunk stripe looks awesome on Bonnie Raitt and probably does on you, too!