Constant blinking and stuttering are behaviours common to those suffering from anxiety. This may or may not be the case in your child's situation. I might suggest you google the phrase "anxiety behaviors in children" or "school anxiety in young children" or similar words/phrases to see if the descriptions remind you of your child. If so, then you should visit your family doctor re advice and/or treatment. All the best ....
blinking eyes while talking is a secondary characteristic of stuttering which could mean that the stuttering is getting more severe. I would suggest seeing a speech-language pathologist as soon as possible. Stuttering is an odd disorder but when caught early enough can be treated and usually stopped. http://www.asha.org/public/speech/disorders/stuttering.htm has information about stuttering and also you can find a professional in your area.
Stuttering at that age is usually developmental, but the eye blinking is a secondary characteristic which is a sign that it will not go away on it's own. People who stutter will often be "stutterers" their whole life, but an SLP can help with techniques to help subdue the severeness of the stuttering and teach how to cut down on the secondary characteristics.
What state do you live in? Many states cover therapy for free so you may want to look into this option. Make sure you report the eye blinking and any other things you have noticed that happen along with the fluency issue. Do you have anyone in your family (or extended family) who stutters(ed)? It's familial, so that would be a significant piece of information as well.
Contrary to popular belief, stuttering is not a speech disorder, it is simply a reflection of the person being easily prone to anxiety, thus stuttering.
You cannot "eliminate stuttering", however one can lessen his stuttering by adopting anxiety relieving techniques when speaking, as I did.
Please see a speech therapist. I have gone through the same problem and it was a terrible experience. It can potentially make your child's childhood very very rough.
Consult with a capable speech therapist and enroll him in various public speaking classes. Once your child can deliver speeches in front of an audience well, it boosts his confidence and thus lessening his anxiety in his day to day speech.