Tom Ewell, a decorated WWII pilot, is recalled to active duty in the Air Force. He dons his uniform to report for duty, but it's 1953, and the Air Force separated from the Army in 1947. His uniform isn't "AF Blue" - it's Army Air Force GREEN! It's a clever joke, NOT a mistake.
Gregory (Tom Ewell) drives a 1955 LIncoln convertible.
The movie Sandy (Rita Moreno) describes to Gregory (Tom Ewell) is, of course, The Seven Year Itch (1955), in which Ewell co-starred less than a year earlier, with Marilyn Monroe.
Gregory is shown arriving in Hawaii on a United Airlines plane marked as a Douglas DC-7. However, the registration number, which can be seen under the left wing, is N37542. This plane is actually a Douglas DC-6, delivered to the airline in 1950. The DC-7 is basically a slightly stretched DC-6, so possibly the airline thought the public would not notice, and it would be seen as flying "the latest" aircraft. This plane was sold in 1961 and flew for the Mexican government and air force, and was finally scrapped in 1976.
A news release from Eglin Air Force Base, Florida, published in the Playground News, Fort Walton Beach, Florida, on Thursday 20 October 1955 (Vol. 9, No. 89, page 16) states: "Information was received in the Office of Information Services this weekend that the Twentieth Century-Fox film company for 'The Lieutenant Wore Skirts' will not be visiting Eglin Air Force Base as originally announced. The United States Air Force Technical Adviser, Lieutenant Colonel Kathleen McClure said that the change in plans was made by the producer."