Yes, the imagery of the rattlesnake is deeply intertwined with the identity of the cause of inter-colonial unity and by extension, the Patriot cause.
You have to keep in mind that in the 18th century prior to the Articles of Confederation in 1777 (and arguably even until the ratification of the Constitution in 1788) each colony operated more like independent countries with their own charters, laws, currency and government structure. This mentality also extended to society, with most individuals identifying more with the colony and county they were from than as “Americans”.
So the snake really emerged as a powerful image of colonial unity against Crown authority in 1754, with the “Join or Die” woodcut from Benjamin Franklin in the Pennsylvania Gazette. It promoted the idea that the colonies were parts of a single rattlesnake, and that they must either unite against the “tyranny” of the English Crown in the Stamp Act or perish as a political unit.
Due to the unpopularity of the Stamp Act, and Franklin’s influence, the snake became a powerful symbol of colonial defiance, and thus a symbol of all those who defied Crown authority like the Whigs in the 13 colonies.
Considering the fact that most Patriots were of the Whig tendency, it is no wonder it became such a popular symbol with Patriot militias and Continental forces.
Furthermore, Benjamin Franklin wrote at length as to how the rattlesnake was a great symbol of American unity in the Pennsylvania Journal on December 27th, 1755. He highlighted the symbolism behind the snake, its wisdom, presence only in the Americas; it is ever-vigilant and though small, packs a deadly punch.
Likewise, its symbolism was such that Paul Revere included it in a plate in the Massachusetts Spy newspaper all the way in Boston in 1774.
Sources:
Benjamin Franklin. “Join or Die”. Pennsylvania Gazette (Philadelphia), May 9, 1754. Newspaper. Serial and Government Publications Division, Library of Congress (1)
Paul Revere. "Unite or Die". Massachusetts Spy (Boston), July 7, 1774. Newspaper. Serial and Government Publications Division, Library of Congress (2)
Margolin, Victor. "Rebellion, Reform, and Revolution: American Graphic Design for Social Change". Design Issues Vol. 5, No. 1, 1988