Thank you for coming to JustAnswer with your question. The SpeechEasy device does not have much information "out there" except for sites that mostly advertise it.
However, there is one listing: Speech-Language Pathology/Stuttering/SpeechEasy
It does not look promising. The small unofficial study that was done showed that more than 50% of the people using the device over a long period of time actually ended up with worse stuttering. There are a couple of references to articles within this article.
Here is the link from the makers of SpeechEasy It really doesn't say alot about how it actually works.
Just in case you haven't found this, here is a link to *Stuttering in Children*. Its a good article with many more links to helpful sites and articles.
Being the mother of a stuttering child (my son is now 28 and rarely stutters anymore), I know this can be terribly frustrating for the child and for the people who love him. You always feel like there should be more you could do to help him. But, according to my son who now has children of his own, the best thing we did was make sure he got speech therapy when he was young. Speech therapists will train the child to change his behavior so that he can control his speech. It takes time and persistence, but it is effective.
My best wishes to you and yours.