An excerpt from the book Self-Therapy for the Stutterer
If you are like many stutterers, you probably do not look people squarely in the eye when you talk to them. Chances are that if you observe yourself carefully, you will find that you usually avert your eyes, particularly when you are stuttering or anticipating a block. And by doing so, you tend to increase any feelings of shame or embarrassment you may have about your difficulty.
Maintaining eye contact will not of itself stop your stuttering, but it will help reduce feelings of shyness and tend to build self-confidence. It is this sensitivity which generates much of the tension which causes or aggravates your trouble. So this guideline calls for you to establish the habit of eye contact with your listener.1,2 
This doesn’t mean that you need to stare fixedly at the person to whom you are talking, but still you should look the other person squarely in the eye more or less continuously. Establish eye contact before you begin to speak and continue to do so in a natural way. Particularly, do your best not to look away when you stutter or expect to.
It is possible that you already practice good eye contact, but more probably you are embarrassed and do not. Remember, it is difficult to observe yourself so do your best to be honest with yourself. You might ask someone with whom you converse, such as a member of your family, to watch and find out if you shift your eyes just before or when you stutter.
Perhaps you look away because you are afraid that your listener will react with pity, rejection or impatience. This is not apt to be true. Using eye contact will enable you to test the validity of your fears, and it should put your listener more at ease. Moreover, by maintaining eye contact you can demonstrate that you are accepting—not rejecting—your stuttering as a problem to be solved.3 When you look away, you are denying the problem.
Anyway, do your best to maintain good eye contact as a habit. You will feel better for doing so, as it will help you combat feelings of inferiority and self-consciousness. Therapists recommend its use in trying to help people who are shy and bashful. Interpersonal communication is always facilitated by eye contact, even if you don’t stutter.4 Good speakers use it naturally.
It is unnecessary to turn or hang your head in shame which may be what you are doing unconsciously when you avert your eyes. We hope you can develop a feeling of self-confidence that you are as good as the next person. Do your best to look the world squarely in the eye.
How to Go About Maintaining Eye Contact
Following through on this rule may represent more of a problem than you think. Many stutterers have become so shy that it is difficult for them to look anybody straight in the eye when they are stuttering. It is suggested that you double-check yourself carefully as you try the following 
Start by looking at yourself in the mirror when alone and faking an easy block. Do you keep eye contact with yourself or do you avert your eyes? Try this repeatedly, making sure that you don’t look away. Then do it when making a severe block. If you find you do not keep eye contact before and during the block, work at it until you find that you can and continue doiong it.
Then make some phone calls looking at yourself in the mirror while you are having real blocks. Watch yourself until you can talk without shifting your eyes during five or more real stutterings.5 To complete this program successfully, this is a necessary step.
As you become more sure of yourself, it will be easier for you to maintain eye contact while talking in general conversation.6 This does not mean that you have to stare fixedly or glare at your listeners, but look at them in a normal, natural way, and though they look away, continue to keep contact.
While talking to others, collect one, two, and then three occasions in which you maintain good eye contact as you are stuttering. Then make it ten occasions.
To prove that you have followed through, it is suggested you write down the names and eye colors of ten people with whom you have stuttered, or write down ten or more words on which you stuttered without losing normal, natural eye contact.
Use your ingenuity in devising other pertinent assignments. Build confidence in your ability to speak with good natural eye contact on all occasions from now on, and you will feel better for doing so. It will give you satisfaction to know that you can comply with this rule and will make you a more effective conversationalist.

1 You must acquire the ability to keep good eye contact with your listener throughout your moment of stuttering. (Van Riper)

2 Develop eye control with your audience and create a friendly atmosphere. (Barbara)

3 The value of eye contact is the effect it has on the stutterer. It almost forces him to keep the stuttering going forward through the word. It’s an assertive behavior and a positive act. It’s hard to withdraw and back off if you are holding eye contact. (Starbuck)

4 Try to maintain eye contact with your listeners. Looking away severs the communication link with your audience and convinces them that you are ashamed and disgusted with the way you talk. (Moses)

5 Read a sentence, look up into the mirror, paraphrase the sentence while maintaining eye contact. (Adler)

6 Be sure you don’t look down or away at the moment of stuttering. Some people will look away no matter how much you try to keep contact. To succeed it is sufficient that you look at them. (Sheehan)