Based on the information you've provided, I don't see any other possibility outside of the fact that you're self-employed.
I invoice them quarterly for the work done
So you're not an hourly wage or salaried employee. They only pay what you charge them. You're self employed as far as the IRS is concerned. Additionally, the self-employment is further supported by the fact that you control how the desired end result of your work is achieved. Those who pay you don't direct you on the who, where, where, what, and how aspects of what you do.
An employee does not use SCH C. An employee gets a W-2 or other equivalent document (such as a T-4 from Canada) if you are true an employee. Only a person who owns there own business and they are the only one who owns that business, uses SCH C. That's you without a doubt.
See the IRS page "How to Report Wage Income Paid by Foreign Governments or International Organizations"?
That doesn't apply to you. You are not an employee, and therefore are not paid a wage. You are paid a charge that you establish based on the work done.
Do I pay Social Security/Medicare Taxes?
Yes, because regardless of who you worked for or who paid you, you physically earned that money while physically in the U.S. It's not "foreign earned" income, because you did not earn it while physically outside of the U.S. This also means that income is not excludable as foreign earned income, unless you paid taxes on that money to a foreign taxing authority or entity. I get the impression that you did not.