I have to give one of my girls eye drops twice daily..for the rest of her life, I have also had several with eye poke etc... so we have got used to it. I will try to explain how I do it.
Firstly, you need to be the boss and dont take any nonsense. they may be small but they can be very bossy and feisty and they need to know that this wont work!
Secondly, I hold her on a surface at just about waist height (I stand up and a table top may work for you?) put him on something he feels secure on and wont slip, he wont panic then.
Then hold him firmly against your tummy/waist with the eye to be treated the nearest to your body. With your outside hand put your finger/s on top of his head just on his forehead, gently (to prevent head going up). The finger on the head is really there as a guide, if you press too hard he will put his head up, so just be gentle. Its just to say,'keep your head still please'.
With your thumb (on the same hand) gently pull the upper eye lid up. Then drop the drop into the eye. I made the mistake of putting the nostle/dropper too close to the eye which did'nt help. Now I hold it a little higher and let the drop fall onto the eye ball. I found I panicked otherwise about the eye ball hitting the dropper! Then close the eye gently.
Its easier said than done but my girl no longer complains, she occasionally tries but I just hold her firmly and say No and she gives up instantly. I treat afterwards helps a lot too! Now I am used to it the others dont ever flinch. You just have to be confident otherwise they sense you getting nervous and will fidget. I describe the piggy agaist your body as the 'piggy headlock'! its not really but I find you have much more control and leverage.
Hope that helps! Just ask if its not clear, its hard to put in words to the process but this is what I find works for me.