A "pink eye" is actually inflammation of the conjuctiva, which is the outermost layer of the eye. This can be caused by a variety of things, including viruses and bacteria, and allergic reactions. The most common form of pink eye, the one seen in children, is caused by a virus and is spread by hand to eye contact. Many bacteria can cause conjuctivitis. The common causes are the bacteria that cause gonorrhea and chlamydia; this can occur in infants (via passage through the birth canal) and in adults (via direct inoculation).

I suppose that farting into someone's pillow might incite an allergic reaction that would look like a pink eye, but in that case the cause would be body's own immune system (the cause of all allergic reactions) rather than a virus or bacterium.

I suppose it is possible however unlikely. 80% of the matter contained in the large intestine is pure bacteria. It has it's digestive purpose there and the body contains it well in that area. bacterial pink eye is the most annoying kind with green or yellow discharges. If bacteria originating in the colon reached and survived on the pillow.. why not?

I would think it probably wouldn't be. Pink eye is a bacterial infection, and while it's certainly true that there are a lot of bacteria around the anus, in order to give something to someone, you would have to be farting on their pillow nude, and they'd have to introduce the bacteria (by touching the pillow and then their eye, or by laying on the pillow) within a few minutes, before it died. I don't know what type of bacteria causes pink eye, but it would have to be ingested, and I don't believe it would survive a trip through the digestive tract.