The "Men In Black" are named Uno (played by John Sayles) and Dos (played by David Strathairn). Although they are not named in the film or end credits, the names were revealed in a photo caption on page 39 of Starlog magazine (#86, September 1984) as part of an interview with director Sayles.
The entrance and exit scenes of the Men in Black's first appearance in the bar were filmed entirely in reverse with the camera upside down, according to director Sayles' commentary on the restored print. The two men started on the bar stools, then backed out of the room for the entrance; while the exit was filmed with the two outside the bar and backing into it, including Strathairn replacing the cookies in the cookie bowl.
John Sayles financed most of this movie himself. According to Wikipedia, "Sayles describes this movie as being about the immigrant experience of assimilation. He spent part of his MacArthur Fellows 'genius' grant on the film, which cost $350,000 to produce".
John Sayles stated that in choosing the steel drum for the music on this movie, with its unusual and technological, yet still organic, quality, he was partly influenced by the zither music from Carol Reed's The Third Man (1949).