Box braids will not help hair grow, but simply help with retention and breakage. When hair is braided, there is less of a chance for the hair to break off.
When hair is braided in any style, there is less of a chance that it will be manipulated into a hairstyle. By styling hair and manipulating it constantly through mechanical means, dead ends from hair breakage are promoted.
This can be prevented in the long run by conditioning hair while it is in box braids, and also not immediately box braiding the hair once it is taken out of box braids. Box braids should be kept in no longer than 2 months at a time. Hair still grows normally as if it wasn't braided; however, the normal human loses 50 hairs a day, not including extra hairs that break off or are pulled out from hair manipulation.
Moisturizing the hair with deep conditioners and regular oil treatments while box braids are in will help the hair to stay moisturized and will also lessen any chances of it breaking off when the braids are removed. By washing the scalp on a regular basis, such as two to three times a month, and applying oils, the moisture content of the hair keeps the strands from becoming brittle.