As cool as that would be, there's no record of the Puckle gun being used by either side. From the Long 18th Century Blog Series:

"Although John Montagu, 2nd Duke of Montagu (1690-1749) purchased several pro-type models of the gun in 1722, the British army and Royal Navy never placed a major production order. Consequently, no historical record of the Puckle Gun fired in combat exists."

James Puckle actually demonstrated the weapon to the English Board of Ordinance several times: in 1717, while the design was still being developed, several times between 1719 and 1721, and, famously, at a public trial in 1722. The gun impressed the audience by firing 63 shots in seven minutes in a heavy rain. However, it seems to have been treated as a hyper-specialized novelty by both its audience and its inventor. Puckle advertised its speed and rapid-fire ability, and added in the same breath that it could shoot square bullets at the Turks (while round bullets would be reserved for Christians) (Willbanks 2004).

Sources:

Michael Burgan, Weapons, Gear, and Uniforms of the American Revolution. Capstone Press, 2012. George D. Moller, American Military Shoulder Arms, Vol. 1: Colonial and Revolutionary War Arms. University of New Mexico Press, 2011.

George C. Neumann, Battle Weapons of the American Revolution. Andrew Mowbray Inc, 2011.

James H. Willbanks. Machine Guns: An Illustrated History of Their Impact. ABC-CLIO, Weapons & Warfare Series, 2004.