vereato says:
WHERE  @User_Name= '" & TxtBox_User_Auth.Text &"'
Here is the issue. You are setting parameter @User_Name in left side instead of column name.
Private Sub Update()
    Dim ConnStr As String = ConfigurationManager.ConnectionStrings("MySQLConnect").ConnectionString
    Using Conn As New MySqlConnection(ConnStr)
        'Using Cmd As New MySqlCommand
        Using Cmd As New MySqlCommand("UPDATE tbl_users SET Register_Time=@Register_Time, Last_Login_Date=@Last_Login_Date, PC_Name=@PC_Name, OS_Version=@OS_Version, IP_Address=@IP_Address WHERE User_Name=@User_Name")
            Using Da As New MySqlDataAdapter()
                Cmd.Parameters.AddWithValue("@Register_Time", DateTime.Now.ToString("T"))
                Cmd.Parameters.AddWithValue("@Last_Login_Date", DateTime.Now.ToString("U"))
                Cmd.Parameters.AddWithValue("@PC_Name", System.Environment.MachineName.ToString)
                Cmd.Parameters.AddWithValue("@OS_Version", System.Environment.OSVersion.ToString)
                Cmd.Parameters.AddWithValue("@IP_Address", Lbl_IP_Address.Text)
                Cmd.Parameters.AddWithValue("@User_Name", TxtBox_User_Auth.Text)
                Cmd.Connection = Conn
                ' Connect to Database
                Conn.Open()
                Cmd.ExecuteNonQuery()
                'Conn.Dispose()
                Conn.Close()
                Cmd.Dispose()
            End Using
        End Using
    End Using
End Sub