Conjunctivitis is common among House Finches. The infected bird has, unfortunately, a very short life expectancy...it will starve if a cat doesn't get it first.
It is highly contagious. If the bird in question is mingling with other finches at your feeder, take it down and clean it thoroughly.
I'm not sure if it can be communicated to humans, but I'd take great care ( rubber gloves, mask ) when dealing with the bird.
What is not forbidden is inevitable.