Is there any evidence that overbreathing can cause adenoid hypertrophy and/or can breath re training shrink the adenoid?

Complex question. If you mean overbreathing through the nose, by the time air reaches the adenoid, it has been cleansed of most contaminants and is at body temperature. Therefore the act of nasal inhalation would have no effect on the size of the adenoid in children. The adenoid is a lympho-epithelial structure that usually atrophies with age. I do not think that any training would effect the size of adenoid.