Change code to require a Response

You forgot to keep in the "Exit Sub" before the "End If". Also, it's great that you created all of those variables. If you want to continue using them you can, but it could also be setup like the following (also added a check to ensure there was a valid date in F2):

Sub NameIt()
With Sheets("Log In Screen")
    If .Range("F2") = "" Or .Range("D6") = "" Or .Range("I6") = "" Then
        MsgBox "You haven't filled in the Date, MEDIC# and/or Crew yet."
        Exit Sub
    ElseIf Not IsDate(.Range("F2").Value) Then
        MsgBox "The Date (F2) is not valid."
        Exit Sub
    Else
        ActiveWorkbook.SaveAs Filename:= _
        "C:\Users\Brian\Desktop\Excell Test Folder\" & _
        Format(.Range("F2").Value, "m-d-yyyy") & "-" & _
        CStr(.Range("I6").Value) & "-" & CStr(.Range("D6").Value) & ".xlsm", _
        FileFormat:=xlOpenXMLWorkbookMacroEnabled
    End If
End With
End Sub