Born in Boston, Massachusetts, on 19 October 1747, Henry Jackson was the youngest son of Joseph and Susannah (Gray) Jackson. Before the War for Independence, he was an officer of the First Corps of Cadets in Boston, which was disbanded during the British occupation. After the evacuation, six former cadet officers organized a company of seventy-eight officers and men called the Boston Independent Company on 17 March 1776, with Henry Jackson as their commander. In January 1777, the unit was taken into Continental service, designated the 16th Massachusetts Regiment, with Jackson at the head with the rank of Colonel. He led his regiment in the Philadelphia campaign of 1777, at Monmouth and Rhode Island in 1778, and at Springfield, New Jersey in 1780.

Jackson's Additional Continental regiment was disbanded in 1781 and Jackson was transferred to command the 4th Massachusetts Regiment. He was promoted to Brevet Brigadier-General, 30 September 1783 and led Continental forces into New York City on the heels of the British evacuation in November. He was retained as Commander the 1st American Regiment (1783-1784), which was the only infantry unit in commission after the dissolution of the Continental Army. Jackson left the regular service on 20 June 1784 when the standing army had been reduced to a handful of soldiers.

He later served as a Major General in the Massachusetts militia from 1792 until 1796 and was the agent supervising the building of the frigate Constitution at Boston in 1797.

He was a lifelong close friend of Major-General Henry Knox, and acted as his agent in his business affairs (particularly those concerning Knox's vast land holdings in Maine) while the General was Secretary of War. Other close associates included Hepzibah Swan of Boston, in whose household Jackson lived for some years.

Henry was an original member of the Massachusetts Society of the Cincinnati from 1783 until his death in 1809. Unmarried, he died in Boston on 4 January 1809.