What are the five most significant events in 1776 by David McCollough, not focusing only on the military engagements?

1776 focuses primarily on military events because the author covered political events of the same year in a previous book: John Adams.
The first important event described in the war actually took place in October 1775, in London. King George III gave an important speech to British Parliament in which he promised to crush the rebellion. After that speech, peaceful reconciliation was out of the question.
The second key event happened in March 1776. The Americans put cannons in Dorchester Heights, forcing the British to evacuate Boston. Henry Knox, head of artillery, had dragged the cannons 300 miles from Ticonderoga to Boston.
The third major event, which the author describes as the most important event in the book, was George Washington's escape from Brooklyn in August. Despite being defeated in battle, Washington managed to extricate his weakened army. Had the British managed to trap Washington, the war would have been over.
A fourth significant event was at the end of 1776. The British drove Washington's depleted forces across New Jersey. With Washington on the verge of defeat, the British went into winter quarters. This enabled the Americans to rally and win victories in the Trenton-Princeton campaign. For the second time, the British had allowed Washington to recover and escape.
A final event that unfolded gradually throughout the year was George Washington's development as a field commander. He learned from the mistakes he made and became much more decisive and capable in subsequent years.
These are the five most significant events in 1776.

It is, of course, difficult to point to exclusively non-military events in a book that focuses so much on the Revolutionary War. So there are some events in this answer that are basically military. Still, some are more political in nature. 
The first event actually occurs in October of 1775. After a speech by King George and a lengthy debate in Parliament, with significant dissenting voices, both houses of Parliament voted to approve Lord North's plan to send a large military and naval force to the colonies, then in a state of all-out rebellion. This marked, in many ways, a point of no return in the conflict and was thus a major event. Another occurred in March of 1776, when Washington's capture and fortification of Dorchester Heights forced the British army to abandon the city of Boston. McCullough describes this event as the "first thrilling news of the war" for the rebels (108). 
Another event, of course, was the decision of the Second Continental Congress to declare independence from Great Britain. This decision, reached on July 2, meant that the war would be an "all-out war for an independent America" and was a major turning point described in the book (137). But this occurred even as another crucial event was taking place--the massive invasion of New York by the British army and navy. Finally, the book closes with a speech by King George in 1777. Echoing the debates in Parliament at the beginning of the book, the speech announced that, despite the previous year's events, the Americans had not "recovered from their delusion" (292).