It sounds like allergic conjunctivitis or viral conjunctivitis.
You can get some eye saline flush at the drug store. Just flush both eye thoroughly with the saline liquid.
You can also try human antihistamine eye drops. You can try the "zyrtec" brand eye drops for allergies. They sell them at all the grocery stores and drug stores now. Put one drop in each eye twice a day as needed.
Keep using the oral antihistamine as well. You can use it long term without problems.
If it is allergic conjunctivitis, you need to keep treating him until allergy season is over.
If it is viral conjunctivitis, it should resolve in 7-10 days.
Conjunctivitis is inflammation of the pink fleshy tissue that is under the eye lids.
If the drainage turns yellow-green, then he should see a vet because that means he has a bacterial infection and needs an antibiotics.
If the white drainage does not respond to symptomatic treatment, he should see a vet because he could have blocked tear ducts or he could have a problem with his tear glands where they are not making enough tears.
I hope this helps.