In my opinion, they gave you incorrect information. When you are working for the government (i.e. in the military), your children are natural born citizens. John McCain is a natural born citizen since he was born in Panama while his dad was serving in the Navy there.
The State Department doens’t concur with your opinion, and Congress lacks the authority to redefine the term natural-born citizen.
Actually there was some murkiness with respect to Panama. Looking at State department records: the clarifications were in 1952 for Guam, 1986 Puerto Rico and US Virgin Islands using the term “citizens at birth”, and the State Department noted that these areas are under US jurisdiction.
The Panama Canal Zone was clarified using the term citizen only. According to the State Department, the canal zone was not under the “jurisdiction” of the US.
The state department also states that current law is that both parents must be citizens and at least one a resident of the US to qualify.