Actually, a topical application IN the eye is suggested as a treatment for eye worm, along with an orally administered drop,
but I don' t know of a recommendation for eye infection or irritation.
First, check to see if your chick has mites -- if so, they may be scratching a lot and accidentally scratch their eyes.
Wash the eye with sterile saline. Just plain saline, don't get the contact lens kind with preservative in it. (Hold the eyelid up and drip a few drops in) This will remove loose dust etc. if there is some irritating the eye.
You can buy an opthalmic anitbiotic ointment to use in the eye if it is an eye infection, and also give something like auromycin or duramycin in the water. There was an eye doctor that wrote in here that it is fine to use regular triple antibiotic ointment instead of the $13 a tiny tube opthalmic ointment.
I have treated mine with the regular antibiotic ointment and had good success.
If the eyes are just swollen and they have any discharge from the nose, coughing, or wheezing, you need to treat with an antibiotic in their water. Separate them from your other birds and watch the rest of your flock for developing symptoms.
ETA: Eye worm: The http://www.vetmanual.org/mvm/index.jsp?cfile=htm/bc/202800.htm states:
As a treatment for Mansons eyeworm, a local anesthetic can be applied to the eye, and the worms in the lacrimal sac exposed by lifting the nictitating membrane. A 5% cresol solution (1-2 drops) placed in the lacrimal sac kills the worms immediately. The eye should be irrigated with sterile water immediately to wash out the debris and excess solution. The eyes improve within 48-72 hr and gradually become clear if the destructive process caused by the parasite is not too far advanced.
VetRx is
NOT a cresol solution.