A counterclaim, in rhetoric, is a claim made to rebut a previous claim. Other delegates to the convention had just argued that it was best to tread cautiously, to hold off on preparing for war with Britain. After all, King George III might come through and compromise, making war unnecessary. They were relying on hope and the triumph of diplomacy.
Henry rebuts this claim by pointing out that history tells them if one looks at "the conduct of the British ministry for the last ten years," it cannot possibly "justify those hopes with which gentlemen have been pleased to solace themselves." He says that they have petitioned for their grievances to be dealt with, but the British have responded with "warlike preparations," and since that is the "last articles to which kings resort," it's foolish to assume the crown is seriously considering anything other than oppression of the Colonies through force. "Fleets and armies" deployed by the crown are incompatible with any hope for "love and reconciliation."