Since essential oils and water do not mix, you need a substance that is safe to use in your eyes and that will quickly emulsify the essential oil, allowing water to quickly flush it all away. Use anything with fat in it, which will dilute the oil.
The answer is milk, butter, or cream.
Quickly flush the eye with milk or cream until it stops burning. This should happen almost immediately. At that point, you can flush the eyes with fresh water to remove the milk.
This is helpful to remember especially when using EOs around children who may accidentally rub some into their eyes or face area.