The definition of a "Flag Carrier", although somewhat vague and frequently disputed, is generally accepted to mean: an air carrier that is owned by, or subsidized by, the government of the country of registration, especially for the purposes of flying international routes.  

**Wikipedia:**

 - A flag carrier is a transportation company, such as an airline or
   shipping company, that, being locally registered in a given state,
   enjoys preferential rights or privileges accorded by the government
   for international operations. The term also refers to any carrier
   that is or was owned by a government, even long after their
   privatization when preferential rights or privileges continue.


[**Business Dictionary**][1]:

 - Airline ... registered under the laws of a country whose government
   gives it partial or total monopoly over international routes.


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The US has no carriers in this category. All subsidizing of major domestic and international routes ended with the Airline Deregulation Act of 1978. (note: there are still some small airlines and routes that are subsidized to serve very small communities... If you want to read more about this program, it's called [Essential Air Service][2]).  


  [1]: http://www.businessdictionary.com/definition/flag-carrier.html
  [2]: https://en.wikipedia.org/wiki/Essential_Air_Service