Tried to login myself, everything worked normally. Net::HTTP 6.00 is installed in my system, and I tried downloading 6.03, all tests succeeded: 
$ perl test.pl
Hurrah! Logged in
$ perl apache-https.t
1..8
# Running under perl version 5.014002 for linux
# Current time local: Thu Jul 12 10:31:42 2012
# Current time GMT:   Thu Jul 12 06:31:42 2012
# Using Test.pm version 1.25_02
# ----------------------------
# 200 OK
# Content-Type: message/http
# Date: Thu, 12 Jul 2012 06:31:47 GMT
# Server: Apache/2.4.1 (Unix) OpenSSL/1.0.0g
# Transfer-Encoding: chunked
#
# TRACE /libwww-perl HTTP/1.1
# Host: www.apache.org
# User-Agent: Mozilla/5.0
# Accept-Language: no,en
# Accept: */*
#
ok 1
ok 2
ok 3
ok 4
# ----------------------------
# 200 OK
# Content-Type: message/http
# Date: Thu, 12 Jul 2012 06:31:47 GMT
# Server: Apache/2.4.1 (Unix) OpenSSL/1.0.0g
# Transfer-Encoding: chunked
#
# TRACE /libwww-perl HTTP/1.1
# Host: www.apache.org
# User-Agent: Mozilla/5.0
# Accept-Language: no,en
# Accept: */*
#
ok 5
ok 6
ok 7
ok 8

[download]
Perhaps the problem lies deeper in the system. Probably, it's related to system libssl?
Sorry if my advice was wrong.

In reply to Re^3: Net::Google::AuthSub always fails to log in by aitap
in thread Net::Google::AuthSub always fails to log in by marvell

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.