If our eyes can do it, our skin can do it. Depicted: Frank Sinatra and Emma Stone.
Humans have only one pigment molecule: melanin. Brown, blue and green eyes are caused by different distributions of melanin molecules in the eye, and different proportion of light absorbed and light scattered.
Melanin absorbs light, so the more melanin there is, the less light
will be (reflected back out of the eye). Brown-eyed people have more
melanin, less light.
The opposite is true for people with "blue" eyes. Those with less
melanocytes can't absorb as much light, so more light is reflected
back out of the eye. This is called scattering — and when light is
scattered, it reflects back at shorter wavelengths. On the color
spectrum, shorter light wavelengths correspond with the color — you
guessed it — blue.
This Rayleigh scattering is the same thing which makes the sky look blue. If distribution of melanin in the eye can make blue or green there, it can make it in the skin as well.
There is another way to make green. We normally have bilirubin in our blood. It is a breakdown product of hemoglobin. Jaundice is caused by a buildup of bilirubin, from liver dysfunction or blockage of the bile ducts. Bilirubin is a sickly yellow color but biliverdin, a precursor product, is a vivid green. You can see this green in an old bruise: blood trapped beneath the skin goes from a dark purple to a vivid green to a mustard yellow as the hemoglobin breaks down.
Higher levels of heme breakdown products in the blood are not in themselves bad for you. A mutant which accumulated biliverdin in the blood and only slowly broke it down to bilirubin would have skin (and sclera) the vivid green of this pigment.
I found a described case of exactly this. This man turned bright green (as opposed to the typical yellow) when he became ill with end stage cirrhosis. His urine was also bright green.
Biomedical Scientist p362 June 2011
Results of the investigation suggested that green jaundice in this
patient was the result of reduced biliverdin reductase activity that
converts biliverdin to bilirubin, perhaps due to a defect in the gene
that codes for the enzyme. Analysis of DNA from the patient’s blood
cells and subsequent sequencing of his bilirubin reductase gene
provided confirmation that this was indeed the case.