Even though they're loosely related, you've presented two separate questions here.
- Why is the call failing when I use the WebBrowsercontrol to load a page via HTTPS?
- Why is the call failing when I use the DownloadFile()method to download a file via HTTPS?
First, you need to eliminate the possibility that your code is failing. Try both of the tasks above using public HTTPS URLs that are known to work correctly.
If you discover that the source of the problem is your private URL, you may want to consider whether you want to ignore SSL errors in your WebBrowser control.
You can do so using the (untested, translated to VB) code from this blog post:
Partial Public Class Form1
  Inherits Form
  Private WithEvents WebBrowser As New WebBrowser
  Private Sub WebBrowser_DocumentCompleted(Sender As Object, e As WebBrowserDocumentCompletedEventArgs) Handles WebBrowser.DocumentCompleted
    If e.Url.ToString() = "about:blank" Then
      'create a certificate mismatch
      WebBrowser.Navigate("https://74.125.225.229/")
    End If
  End Sub
End Class
<Guid("6D5140C1-7436-11CE-8034-00AA006009FA")>
<InterfaceType(ComInterfaceType.InterfaceIsIUnknown)>
<ComImport>
Public Interface UCOMIServiceProvider
  <PreserveSig>
  Function QueryService(<[In]> ByRef guidService As Guid, <[In]> ByRef riid As Guid, <Out> ByRef ppvObject As IntPtr) As <MarshalAs(UnmanagedType.I4)> Integer
End Interface
<ComImport>
<ComVisible(True)>
<Guid("79eac9d5-bafa-11ce-8c82-00aa004ba90b")>
<InterfaceTypeAttribute(ComInterfaceType.InterfaceIsIUnknown)>
Public Interface IWindowForBindingUI
  <PreserveSig>
  Function GetWindow(<[In]> ByRef rguidReason As Guid, <[In], Out> ByRef phwnd As IntPtr) As <MarshalAs(UnmanagedType.I4)> Integer
End Interface
<ComImport>
<ComVisible(True)>
<Guid("79eac9d7-bafa-11ce-8c82-00aa004ba90b")>
<InterfaceTypeAttribute(ComInterfaceType.InterfaceIsIUnknown)>
Public Interface IHttpSecurity
  'derived from IWindowForBindingUI
  <PreserveSig>
  Function GetWindow(<[In]> ByRef rguidReason As Guid, <[In], Out> ByRef phwnd As IntPtr) As <MarshalAs(UnmanagedType.I4)> Integer
  <PreserveSig>
  Function OnSecurityProblem(<[In], MarshalAs(UnmanagedType.U4)> dwProblem As UInteger) As Integer
End Interface
Public Class MyWebBrowser
  Inherits WebBrowser
  Public Shared IID_IHttpSecurity As New Guid("79eac9d7-bafa-11ce-8c82-00aa004ba90b")
  Public Shared IID_IWindowForBindingUI As New Guid("79eac9d5-bafa-11ce-8c82-00aa004ba90b")
  Public Const S_OK As Integer = 0
  Public Const S_FALSE As Integer = 1
  Public Const E_NOINTERFACE As Integer = &H80004002
  Public Const RPC_E_RETRY As Integer = &H80010109
  Protected Overrides Function CreateWebBrowserSiteBase() As WebBrowserSiteBase
    Return New MyWebBrowserSite(Me)
  End Function
  Private Class MyWebBrowserSite
    Inherits WebBrowserSite
    Implements UCOMIServiceProvider
    Implements IHttpSecurity
    Implements IWindowForBindingUI
    Private myWebBrowser As MyWebBrowser
    Public Sub New(myWebBrowser As MyWebBrowser)
      MyBase.New(myWebBrowser)
      Me.myWebBrowser = myWebBrowser
    End Sub
    Public Function QueryService(ByRef guidService As Guid, ByRef riid As Guid, ByRef ppvObject As IntPtr) As Integer Implements UCOMIServiceProvider.QueryService
      If riid = IID_IHttpSecurity Then
        ppvObject = Marshal.GetComInterfaceForObject(Me, GetType(IHttpSecurity))
        Return S_OK
      End If
      If riid = IID_IWindowForBindingUI Then
        ppvObject = Marshal.GetComInterfaceForObject(Me, GetType(IWindowForBindingUI))
        Return S_OK
      End If
      ppvObject = IntPtr.Zero
      Return E_NOINTERFACE
    End Function
    Public Function GetWindow(ByRef rguidReason As Guid, ByRef phwnd As IntPtr) As Integer Implements IHttpSecurity.GetWindow, IWindowForBindingUI.GetWindow
      If rguidReason = IID_IHttpSecurity OrElse rguidReason = IID_IWindowForBindingUI Then
        phwnd = myWebBrowser.Handle
        Return S_OK
      Else
        phwnd = IntPtr.Zero
        Return S_FALSE
      End If
    End Function
    Public Function OnSecurityProblem(dwProblem As UInteger) As Integer Implements IHttpSecurity.OnSecurityProblem
      'ignore errors
      'undocumented return code, does not work on IE6
      Return S_OK
    End Function
  End Class
End Class
Regarding problem #2: It appears you may be confusing WebBrowser and DownloadFile(). As you've probably already discovered, the WebBrowser control doesn't download files. However, you can simulate the behavior using this technique:
Partial Public Class Form2
  Inherits Form
  Private Sub WebBrowser_Navigating(Sender As Object, e As WebBrowserNavigatingEventArgs) Handles WebBrowser.Navigating
    Dim sFilePath As String
    Dim oClient As Net.WebClient
    ' This can be any conditional criteria you wish '
    If (e.Url.Segments(e.Url.Segments.Length - 1).EndsWith(".pdf")) Then
      SaveFileDialog.FileName = e.Url.Segments(e.Url.Segments.Length - 1)
      e.Cancel = True
      If SaveFileDialog.ShowDialog() = DialogResult.OK Then
        sFilePath = SaveFileDialog.FileName
        oClient = New Net.WebClient
        AddHandler oClient.DownloadFileCompleted, New AsyncCompletedEventHandler(AddressOf DownloadFileCompleted)
        oClient.DownloadFileAsync(e.Url, sFilePath)
      End If
    End If
  End Sub
  Private Sub DownloadFileCompleted(Sender As Object, e As AsyncCompletedEventArgs)
    MessageBox.Show("File downloaded")
  End Sub
  Private WithEvents SaveFileDialog As New SaveFileDialog
  Private WithEvents WebBrowser As New WebBrowser
End Class
In any event, the first step in solving this is to figure out whether it's your code or the private URL that's causing your issue.