The WinZipSub subroutine plays a central role. This routine uses Shell to start WinZip32.exe, passing it -a to archive files or -e to extract files. The cmdWinZip_Click and cmdUnWinZip_Click event handlers call this routine to zip and unzip files.

' Zip or unzip the files.
' The default location of the winzip32.exe is
' C:\Program Files\WinZip\. If your winzip32.exe
' is in another folder, change ZIP_EXE accordingly.
Sub WinZipSub(ByVal source As String, ByVal target As _
    String, ByVal zip_it As Boolean)
Const ZIP_EXE As String = """C:\Program " & _
    "Files\WinZip\winzip32"""

    If zip_it = True Then
        ' Zip appending the source file to the target.
        Shell ZIP_EXE & " -a " & target & " " & source, _
            vbHide
    Else
        ' Extract the target to the source directory.
        Shell ZIP_EXE & " -e " & target & " " & source, _
            vbHide
    End If
End Sub

Subroutine cmdViewUnzip_Click uses Shell Execute to "open" the temp directory. That invokes the directory's default action which is to display the directory's contents.

' Zip up the files in the current directory.
Private Sub cmdWinZip_Click()
Dim next_file As String
Dim target As String
Dim source As String

    ' Loop over all the files in the current directory.
    next_file = Dir(m_AppPath & "*.*", vbNormal)
    Do While next_file <> ""
        source = """" & m_AppPath & next_file & """"
        target = """" & m_AppPath & "temp\Project.zip" & _
            """"

        ' Add this file to Project.zip
        WinZipSub source, target, True
        next_file = Dir
    Loop

    cmdViewWinZip.Visible = True
End Sub

' Open the zip file.
Private Sub cmdViewWinZip_Click()
    ' Open the Zip file.
    ShellExecute 0, "Open", """" & m_AppPath & _
        "temp\Project.zip" & """", vbNullString, _
        vbNullString, 1
    cmdUnWinZip.Visible = True
End Sub

' Unzip the zip file.
Private Sub cmdUnWinZip_Click()
Dim target As String
Dim source As String

    ' Compose the source (destination directory)
    ' and target (Zip file) names.
    source = """" & m_TempPath & """"
    target = """" & m_AppPath & "temp\Project.zip" & """"

    ' Extract the files.
    WinZipSub source, target, False

    cmdViewUnzip.Visible = True
End Sub

' Open the temp directory to display its files.
Private Sub cmdViewUnzip_Click()
    ' Open the directory.
    ShellExecute 0, "Open", """" & m_TempPath & """", _
        vbNullString, vbNullString, 1
End Sub

' Create the temp directory.
Private Sub Form_Load()
    ' Get the application and temp directories.
    m_AppPath = App.Path
    If Right$(m_AppPath, 1) <> "\" Then m_AppPath = _
        m_AppPath & "\"
    m_TempPath = m_AppPath & "temp\"

    ' Make the temp directory if it doesn't exist.
    If Dir(m_TempPath, vbDirectory) = "" _
        Then MkDir m_TempPath
End Sub

' Remove the temp directory.
Private Sub Form_Unload(Cancel As Integer)
Dim next_file As String

    ' Remove the files in the directory.
    next_file = Dir(m_TempPath & "*.*", vbNormal)
    Do While next_file <> ""
        Kill m_TempPath & next_file
        next_file = Dir
    Loop

    ' Remove the directory.
    RmDir m_TempPath
End Sub

Thanks to Pooya Kashefifar for pointing this out!

WinZip is such a worthwhile program that I encourage you to register.