I think you should go for a fiery red shade like the ones below...

Green sounds like the obvious color, but I dunno green sounds kinda blah. How about purple? It's not the perfect color for green eyes (I like reddish brown or auburn for that), but since you want an unnatural color, that's what I'd pick. You could also just go with black or brown with purple hilites. To tie in your hair and eyes, try some purple eyeliner.

I know I was down on green, maybe because I tried it once and I looked awful. But I found a few pictures of green hair that are edgy and cute and would be aesome with green eyes.

I think red could bring out the most of your green eyes..I also have the same eye color and I rock a chic red hair color...

Check out my favorite hair color shades...

If you were in Chicago, I would dye it a deep dark red

I have the same reddish face problem, but I still dyed my hair pink. I had a very spontaneous hair dying experience, the only thing I would recommend is being completely happy with the color, then you won't regret it because of your face. Honestly people will probably look at your hair more than paying attention to the undertones of your face. Pick a color that brings out your eyes, make them the main focus of your face! Rock what you got girl!!!