Another thing I should add is that if the injury is caused by a chemical burn, you should not cover the injuries of either eye if possible until it is thoroughly irrigated. Covering a chemical burn on an eye can trap the chemical and make it worse and even spread it from one eye to the other if the same dressing is applied across both eyes.
In regards to an eyeball injury, if one eye is injured, you should cover both eyes if possible. Leaving one uncovered will lead to you moving both eyes, obviously, which will cause more damage. If you are by yourself and need to move this won’t work. If you have someone to transport you and get you to medical attention (an emergency department), then covering both eyes is best practice.