As far as i know, it won't cause pink eyes. Pink eye is the inflammation of the thin, clear covering of the white of the eye and the inside of the eyelids. There are different kinds of pink eye and it can usually bring watery, itchy eyes and sensitivity to light. The treatment of pink eye will depend on the type of pink eye you have, so you need to consult your doctor and get treatment.
Eye drynss syndrome is a very common but annoying problem, referring to deficient tear production or excessive tear evaporation or drainage. Tear works as lubrication, nourishment and anti-bacterial agent in eyes. Without enough anti-bacterial agent in eyes, it makes easier to get bacterial infection. So it is possible to have pink eye because of long term eye dryness. And severe dryness might lead to inflammation, scarring on the surface of cornea for lacking of lubracation. To prevent it, you should reduce the dryness. Applying warm compress is helpful to tear production. You can do the warm compress 4-6times per day, 10-15 minutes each time. Make sure the air around is moist enough otherwise your eyes would be drier. Use a humidifier to moisturize the air, which make you feel better and the dryness would be relieved.
Yes, long term of dry eyes will cause pink eyes which could be caused mainly by the infection at the eyes because of coming invisible bacterium. You should use the eye drops to moisture the eyes. In addition, you could use the warm compress to make the eyes get moisture. You could eat more food with vitamin C to moisture the eyes. You could just do the above ways to prevent the dryness of the eyes.