That she is a veteran, because of her military record is not in dispute. I don’t know if she claimed to be a Gulf War Veteran or if Fox claimed she was.
Those of us who served in the Gulf War were in an AO (Area Of Operations) that was strictly defined by the DoD. I’m not sure what it was exactly, but it encompassed the Arabian Peninsula, the Iraqi AO, the Persian Gulf, Arabian Sea and certain other parts of the Middle East and southern Turkey. The award of the Southwest Asia Service Medal during the time of (approx) Aug. 1990 to about April or May of 1992 was given to all US military personnel who served in the AO.
In addition, for some of us who were in the AO from the time of Aug 1990 to Feb-Mar 1991, the dates the buildup and actual shooting began and ended, the Saudi Arabian government awarded all Coalition military personnel in the AO, the Kuwaiti Liberation Medal, which can be worn on a US military uniform. The Government of Kuwait chimed in later an awarded the Kuwaiti Kuwait Liberation Medal to us. This was carried to its most ridiculous extreme, since it closely resembled the Saudi Kuwait Liberation Medal. Since, in the Air Force, we can choose not to wear all of our awards and medals, I promptly awarded my Kuwaiti medal to my dog for faithful service. It hangs on his collar to this day.
Ms. Conger, by her service in Utah and Korea during the time frame ending in 1991 would be awarded the National Defense Service Medal, an additional US award that was given to all US military personnel in that period, regardless of where she served.
As a minimum, those of us that served in the Gulf War and in the Persian Gulf AO received the Southwest Asia Service Medal, the (Saudi) Kuwait Liberation Medal, and the National Defense Service Medal.
All these together are what we referred to as “I was there” medals, and really don’t particuarly make a big impression within the military circles, both active and retired. Consider, that if there were any more US military personnel on the Saudi peninsula just prior to the start of the shooting, we might’ve sunk the country into the Gulf from the sheer weight of all of us and our equipment. 
With these parameters, you make the call. Just among us retired military types, the general rule is that you should’ve been walking, floating or flying in the AO during the actual wartime operations to call yourself a Gulf War vet.
…send lawyers, guns, and money…
Warren Zevon