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?