According to the 1800 US Census, 17% of the US population back then were slaves. This seems rather hard to square with a graph claiming that more than 80% of the labor force were "free agents".

I think Morgan's figures may have excluded slaves. They weren't paid a paycheck either, so they fall in neither category.

Yes, I was suspecting that the underlying data excluded slaves. Not sure I buy the "paycheck" argument --- presumably in the less tamed regions of the US, there were plenty of other people who hardly ever handled money.

Driven in part by soaring health care costs for full-time employees.

That's one of the drivers, certainly, but there's many more forces. Check out Dan Pink's "Free Agent Nation" for more. It's a few years old by now, but still the best view of the overall trend.