Here's very short video showing the technique. It appears a vet is doing it and it really does look easy. I would have some Q-tips handy as well as some soft tissues. Then I'd flush the eye very thoroughly after you're finished.
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Do you have a friend who could come over and give you moral support? Or a friend that's a nurse or nurse's aid? I had a nurse friend come over and help me with a necropsy one time and I not only learned a lot, but it was so much easier with someone there that found it all very normal and interesting. And you will find it easier to have someone hold your hen still, wrapped tightly in a towel to confine her legs and wings while you work on the hen.
The pus will not respond to lancing, and you don't need to do any cutting or puncturing to remove the pus. Did you watch the videos yet? It's just a simple matter of putting pressure at the outer edges of the swelling to force the pus out of the eye. You'll know when all the pus is out when the eye settles back into its normal shape. Your hen is going to feel a lot better after you do it.
Ah! Okay—thank you! I’m at the feed store now and all they have is tylan 50 unfortunately. I don’t know how to get the recommended antibiotics without going to the vet.The eye pretty much "floats" in its orbit. Anything behind the eye, pus or even worms, will pop out with a little pressure on the swollen tissue. It doesn't erupt through the skin as would a pustule on a human.
You should be able to find terramycin eye ointment at the feed store.