They both are correct, but have different meanings.
To "Look someone in the eyes" is looking at them directly.
It is important to look someone in the eyes when you are talking to them.
He walked up to the counter table and looked me in the eyes and said, "This iPod does not work."
To "Look someone in the eye" is to look at them intensely, almost like an inspection.
He looked me in the eye and said, "I love you."
You can also say "Look me in the eye and tell me the truth."
He looked her in the eye and asked, "Did you take that cookie?"
This is asking someone to look at you and be truthful. (Because we know that liars always look to the side or away from you when answering)