White flower oil (made since 1927 and named after it's creator's favorite flower, White Narcissus, rather than its ingredients) is made by creating a combination of a number of oils. It combines wintergreen oil with peppermint, menthol, eucalyptus, camphor, and lavender essential oils. Wintergreen oils should not be put in your eyes but a small accidental dose should result in only temporary irritation. The same goes for peppermint, eucalyptus, and lavender oils. Menthol and camphor oils should not be put into your eyes in undiluted form but some ophthalmic solutions include them as components. Eye injury has been associated with the (accidental) application of Olbas oil which has some of the same ingredients with the notable additions of mint, cajuput, juniper, and clove oils.
In the case of accidental application of any essential oil to your eye(s) you should rinse with milk, as it will bind the oil better than water, and then rinse the milk out with water.
If a small dose gets in your eye, stings a bit, and then you feel better: you're probably ok. However, due to individual sensitivities and allergic reactions, if there's any lingering irritation or worry than seek professional help. It is you're eyesight we're talking about.