There are many reasons why a person may struggle with making eye contact. Some gifted kids who take in a lot of stimulation at a high level find it hard to attend to all that stimulation at once. Breaking eye contact can make it easier for them to listen and to process verbal information.
Also, some people who are more sensitive and emotionally intense may be picking up a lot of information from looking at your face. Most often when parents say "look me in the eye" it is when they are probably communicating about something that is high in emotions and that may be overwhelming to the child. Also, when eye contact has become such a loaded issue she's already no doubt feeling on the defensive every time she realizes you are upset about it.
If you overall have concerns about her development by all means get her evaluated by someone who understands giftedness.
If the eye contact thing is an isolated concern honestly I would totally leave it alone. If it is a stand alone issue when she's older she can learn ways to modify and cope with it to keep up with the social standard. I would very strongly caution against telling her she's rude or making her responsible for your emotions of upset. That's a pretty big burden for a kid to carry over something like this. I'd instead say something like "people feel differently about eye contact, some people feel uncomfortable about it but they learn to handle it as they get older. It does help me when I'm talking to know that you are listening so let me know if you understand what I'm saying..."
