If only one eye was involved, then it's possible that he might have scratched or injured the cornea in some way. While I suppose it's possible that he's injured both eyes, it seems less likely.
It sounds as if he may have uveitis ( http://www.petmd.com/dog/conditions/eyes/c_dg_anterior_uveitis) which is inflammation in the structures of the eye. This condition can be painful which is why the patient squints or prefers to keep their eyes closed. They also tend to be worse in bright light.
The possible underlying cause for this condition is quite varied and can range from viral/bacterial infections to metabolic issues to trauma to glaucoma although the largest percentage of them appear to be Idiopathic...which means we can't find an underlying explanation for the condition.
Most patients will respond to topical anti-inflammatory medication and those which provide pain relief; unfortunately, these drugs would have to be dispensed by a vet.
The prognosis for most patients is good although there can be complications if the condition isn't diagnosed properly or quickly. Therefore, if at all possible, I'd encourage you to have him seen, if you can.
If not (for whatever reason), and he's not vomiting nor currently taking nsaid or steroid medication, then Aspirin may help to provide some relief although it's not likely to provide 100% relief. The dose would be 10 mgs per pound of body weight twice daily, with food to avoid stomach upset.
Just advise your vet that you gave him this medication so that it doesn't interfere with what they might want to prescribe if he has to be seen.
I hope this helps. Deb