In the scene where Alan Shepard is landing aboard the carrier (while imitating Jose Jimenez on the radio), the aircraft seen landing is a McDonnell-Douglas A-4M Skyhawk. There are distinctive features which make it clearly an "M". The "M" version of the A-4 did not enter service until the late 60's/early 70's and to the best of my knowledge was never in service with an operational Navy Carrier Squadron; it was flown (and maintained by guys like me) by the Marines (the Navy having switched to the Vaught A-7 for light attack).