What would Rousseau and Bossuet say about each other’s arguments? (Obviously they would disagree, but what might they say about the other’s lines of reasoning? For example, what would Bossuet think of the idea of equality? What would Rousseau think about royal authority and why? Is there any place where they might agree on something?)

Bossuet was a theorist of divine right monarchy. He believed that the only proper form of government was one that was divinely ordained. Given the nature of the relationship between God and man he saw in Scriptures, he thought a monarch, accountable to none but God, was the only such government. The monarch would be the sovereign—that is, he would represent and embody all the people. The people, on the other hand, would be given order and safety, which Bossuet thought were unattainable in a government where the people had a voice or a claim on liberties. Rousseau, of course, would have disagreed with most of this. He thought that mankind was inherently free but had been repressed by modern states and societies. As he said in The Social Contract:

Man is born free, and is everywhere in chains.

Rousseau, then, thought the only form of government was one which created the maximum possible freedom for its people. He thus advocated something fairly close to a direct democracy, in which the people would come together regularly to express what he called "the general will." So his conception of the foundation of government was radically different from that of Bossuet, who would have strongly objected to his democratic beliefs. But Rousseau thought that the general will commanded absolute obedience from the people, and in this he was not unlike Bossuet, as well as other thinkers like Bodin and Filmer. He believed in an absolute sovereign that had to be obeyed, but for him the sovereign was the people rather than a monarch.
http://www.iupui.edu/~histwhs/H114.dir/H114.webreader/H114.read.a.Bossuet.html