Burrough was born in Underbarrow, Westmorland, and educated in the Church of England, but became a Presbyterian before becoming convinced (a Quaker term referring to conversion).
Upon the Restoration in 1660, Burrough approached King Charles II requesting protection and relief of Quakers in New England, who were being persecuted by Puritans there.
Charles granted him an audience in 1661, and was persuaded to issue a writ stopping (temporarily) the corporal and capital punishments of the Quakers in Massachusetts.
Charles's writ commanded the Massachusetts authorities to send the imprisoned Quakers to England for trial, but they chose instead to release them.
An order for his release signed by Charles II was ignored by the local authorities, and Burrough remained in Newgate until his death on February 14, 1663, aged just 29 ("twelfth month 1662" in the Old Style and Quaker terminologies).