I Cary liquid skin, which is superglue with anti-bacterial properties, and it also doesnt sting like hell on a cut.
The cornea is the fastest healing tissue in the entire body, so there's not much you do for an eye injury other than keep it flushed, and rest.
If you somehow end up with an empaled eye, the usual treatment is to leave the object in place, secure it with a headwrap, and cover the good eye as well. As you move your non injured eye to look around, you're going to move your I jured eye as a reaction even if you don't try to. This can cause further Injury to the impaled eye.
I'm that scenario, I might just try and remove the object, and self extract if I was in the woods, and drive myself to the hospital. Rescue isn't glamorous and will likely consist of poorly trained volunteers who just walk with you out of the woods.