EGIL was an agency or title held by a Starfleet officer at Starfleet Command's Fleet Operations Center, Sol Sector.
In January of 2155, Starfleet officers J. Nygren and W. Mayer were working in or as EGIL. It and the officers' names were listed on directory placards. (ENT: "Demons", directories)
EGIL, like many of the other agencies or titles on the directories, was a reference to NASA and ISS terminology, in this case, the "Electrical Generation and Integrated Lighting Systems Engineer".