Is it true that if I work overseas as a contractor I can pay zero taxes?

Yes and no. If you are self-employed and you live overseas, you may still be required to pay self-employment taxes in the US, including Social Security. This would need to be paid before the Foreign Earned Income Exclusion (which can exclude over $100k of income from US taxation), so it is an actual out-of-pocket expense, not something that would be offset on your expat tax return.

The self-employment tax rate is 15.3%. The rate consists of two parts: 12.4% for social security and 2.9% for Medicare.

However, the US has totalization agreements in place with 25 foreign countries which will affect this tax, and which government your social security contributions (or equivalent) are paid to. Read more on the Social Security website and check whether your country has an agreement in place with the U.S.

The rules are different for each country, so visiting the Social Security website is the best way to see how this will work in your exact host country.