Why are there no purple or green stars?
Stars are dense and opaque spheres of gas in equilibrium, very close to the concept of black bodies defined by Max Planck in 1900. Quantum mechanics tells us that such bodies emit a continuum of light that peaks from red to blue as the temperature increases. The analogy would be from iron metallurgy or your oven, where the metal becomes bright red and then orange as it warms up. To have a blue metal you would need to heat it up to about 10,000 degrees Celsius (18,000 degrees Fahrenheit); this is different to a blue flame which is created by chemical reactions.
Therefore most stars are red, orange, yellow-white or blue. To obtain other colours it is necessary to suppress a large part of the emitted light, which can be done by absorbing atoms at the stellar photosphere. Given the wide variety of stars and surface compositions, there are indeed some examples of unusual colours. For instance, white dwarfs are former stars that have ended the nuclear conversion of helium to carbon. Some of them still have a large amount of carbon at their surface, which strongly absorbs the green in its molecular form. These objects are too faint to be seen with the naked eye, but would appear to have a whitish purple or pink colour because of the missing green light.
Usually it takes molecules to significantly absorb certain colours, which is why planets like Uranus can appear greenish due to methane absorbing the red light. However, this does not
work at hotter temperatures, and we do not know of any green stars.