A hedge school is not some finishing academy for fund managers sponsored by the Titans of Wall Street, or an arcane off-shoot of the forest school movement; rather, it's the name given to an educational practice in 18th and 19th century Ireland, so called because of its rural, illicit nature. Though such schools rarely actually took place behind hedges; metaphorically they did, hidden away from the prying eyes of the English oppressor, and those of their Irish fellow-travellers.
The schools began in the wake of 17th century legislation known as the penal laws which persecuted Irish Catholics and Ulster Presbyterians. These laws restricted the religious, political and economic liberties of Catholics and Dissenters and education was effectively denied to the majority of the Irish population who clung to their traditional language and religion, especially 'beyond the Pale', the area around Dublin. Many middle class Catholics, and the Catholic gentry, went to France to be educated, while this underground network of hedge schools was set up to educate those who could not afford to travel.
In these, local educated men began an oral tradition of teaching the community. Most schools met in private houses and barns rather than in the open as 'hedge' suggests. Unlike today's forest schools, they were emancipatory in nature, and in time came to be tolerated, and the spirit of this tradition is kept alive in modern hedge schools which are usually even farther from a hedge than were the original ones.
I explored this history from a reference in Robert Wyse Jackson's wonderful little book 'Dublin', published by the Eland Press.