The reference Henry makes that "it's coming out in June" is to the movie Star Trek II: The Wrath of Khan which was released on June 4, 1982. He also says "the first one was terrible", referencing Star Trek: The Motion Picture.
As Paige questions why Elizabeth has to leave for work late at night, Henry watches an episode of the TV show Diff'rent Strokes and eats Jiffy Pop popcorn.
The brown 1972 Dodge Coronet Elizabeth drives to pick up Jared is the same car driven by the man who picks up Paige and Henry when they were hitchhiking from the mall.