Add a bit of context, your eyes do not have pain sensors on them, this is not the same as in the back of your eye or other areas, just the front of your eyeball. The pain you feel is 100% from your eyelids, which have quite a few pain and sensation receptors. Most of these are at the central margin.
When you squeeze your eyes shut you are forcing the liquid out from the margin of your eyes, and hopefully out of your eyes entirely. Still, it is no longer in contact with the margin because of the force squeezing it out, so it stops hurting. Start blinking and you have fresh tears which move the soapy solution around, and it gets all over your lid margins and the pain comes back. Repeat until resolved.
Soap does a number of other things like dries your eye out, changes the lipid balance and can make your eyelids hurt for a bunch of other reasons as well. Its fun stuff.
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