500 SSL negotiation failed

I have a new-onset problem on my Windows XP Pro system, demonstrated by the Perl code below (which is, of course, a very cut down example from a much larger program).

It used to work until a few days ago, and I'm pulling my hair out trying to figure out what might have changed on the system to stop it working, and I'm hoping someone here might be able to give me some clues. (It still works fine on my Windows 8.1 system.)

The issue is that the code below (now) fails with "500 SSL negotiation failed".

use strict;
use warnings;

use HTTP::Request;
use LWP::UserAgent;

$ENV{HTTPS_DEBUG} = 1;

my $url = "https://secure.quksdns4.net:2087/";
my $ua  = LWP::UserAgent->new;
my $req = HTTP::Request->new (GET => $url);
my $res = $ua->request($req);
my $sts = $res->code;
my $hdr = $res->headers_as_string;
my $txt = $res->content;

print "\n".$sts."\n\n".$hdr."\n";
print $txt if ($sts == 500);

exit;

The output is:

SSL_connect:before/connect initialization
SSL_connect:SSLv2/v3 write client hello A
SSL_connect:error in SSLv2/v3 read server hello A
SSL_connect:before/connect initialization
SSL_connect:SSLv3 write client hello A
SSL3 alert read:fatal:handshake failure
SSL_connect:failed in SSLv3 read server hello A
SSL_connect:before/connect initialization
SSL_connect:SSLv2 write client hello A
SSL_connect:error in SSLv2 read server hello A

500

Content-Type: text/plain
Client-Date: Sat, 25 Oct 2014 14:52:43 GMT
Client-Warning: Internal response

500 SSL negotiation failed:

Curiously however it works (albeit not very usefully!) if the port number (:2087) is removed.

Active Perl v5.8.8 (which I haven't changed in years), ssleay32 & libeay32 dlls are 0.9.8.1 (also unchanged in years), and while there's a few on the system those in C:\Perl\bin are the only ones in the path.

Any hints as to what might have changed to stop the above working gratefully received!

In short: I guess the peer just disabled SSL 3.0 (at least on port 2087) because of the POODLE attack and because you are still using really old software on an unsupported OS you still attempt to connect with SSL 3.0.

Edit: It looks like version 0.57 of Crypt::SSLeay (needed for LWP at this time) used already SSLv23 handshakes which should in theory be compatible with TLS 1.x. This can also be seen in the debug output (SSLv2/v3 write client hello). So I guess that the reasons might be at least one of the following:

You are using an openssl version without support for TLS1.0. You give as the version number 0.9.8.1, but this kind of version never existed. Either you mean 0.9.8l which looks similar (and supported TLS1.0) or you mean something completely different.
They not only removed SSL 3.0 from the peer, but also straightened the ciphers so that it now requires ciphers which your old OpenSSL does not support yet.
Or they not only require TLS 1.0+ but TLS 1.1+. But support for TLS1.1 is only included since OpenSSL version 1.0.1.