They don't look that way in the sky, but are there stars that look that way up close?
Nope. The reason is simple: stars radiate as black bodies, and black bodies can only take on these colors:
There are of stars whose light peaks in the green or purple part of the spectrum (our Sun peaks in the green), but due to the nature of human vision the star's overall color cannot be green or purple.
Adding to ideasrule's comments:
Stars tend to put out light in a broad spectrum - it spans the visible light spectrum. The curve of emission will tend to be a somewhat smooth dropoff "up-spectrum" and "down-spectrum".
So, even if the star's light peaks in the green part of the spectrum, it still shines in the red and blue. So we see white.
Red stars are red because they peak in the red, and ghave enough room to drop off in the green or blue, AND the dropoff to the down-spectrum isde is in the infrared where was see nothing.
Blue stars are blue because they drop off in the red and green, AND because their up-spectrum is in UV where we cannot see.
That's why you won't see a green star. To see green, you'd have to have a relatively narrow band of emission which drops off before red and before blue. Just doesn't happen.
As an aside, star colors (as seen through high-quality small telescopes) can vary greatly. Ever looked at Alberio (binary) through a really good refractor? The contrast of the rich gold and the blue is amazing. And that's just through human eyes that are notoriously crappy at differentiating colors in weak sources. Digital sensors have revealed a much wider range of star colors in recent years, and yes, stars do peak at some "interesting" frequencies to give some more colors that we are not used to seeing visually.
How many here have spent much time looking at the great Orion nebula? Even in really ideal conditions, the human eye has a hard time registering much more than a pale green. Photographic emulsions and digital sensors show us pinks, reds, blues, purples, etc. Let's not get too anthropomorphic regarding the colors of astronomical objects. We are, after all, collecting very few photons from very distant emitters, and trying to integrate them. Our eyes and brains cannot integrate faint signals over time. Film and digital sensors can. Which is real?
I remember photographing the Orion nebula for the first time and being astonished at the amazing colors my camera managed to register. It wasn't even a particularly capable camera, nor was my telescope particularly good, but the color & detail the photo revealed was much more than what I could see with my naked eyes.
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