So, the answer is probably yes . . .with conditions. I would visit the US Embassy in country in question -- and get their advice -- particularly if the real father will not acknowledge paternity.
US Citizenship for Child Born Out of Wedlock Abroad
Here is the law from US State Department . . .(see Travel.State.Gov site)
Birth Abroad to Two U.S. Citizen Parents in Wedlock: A child born abroad to two U.S. citizen parents acquires U.S. citizenship at birth under section 301(c) of the...
Only in Connecticut
I too recently received a couple of nasty gram letters from Mr. Dodd regarding an unpaid Property Tax bill from the City of Fairfield from 1992. Yes, 1992! (and I left CT in 1996 and have never been back).
Rather than deal with the collections agency, which I would agree...