<<<
GIANT EDIT: The error message originally posted here had a root cause that differed from the problem description. It has now been fixed, but the LWP::UserAgent issue still persists. The issue's root cause appears to be the same: something to do with SSL.
>>>>

Using LWP::Simple, it's refusing to connect to a valid SSL endpoint... I've tried

$ua = LWP::UserAgent->new(ssl_opts => { verify_hostname => 0 });

... and ...

my $ua = LWP::UserAgent->new();
$ua->ssl_opts( SSL_verify_mode => IO::Socket::SSL::SSL_VERIFY_NONE, SSL_hostname => '', verify_hostname => 0 );

... No luck. Any suggestions? Here's what I'm getting back:

WNS Error: HTTP::Response=HASH(0x30e7a30)->content.
$VAR1 = bless( {
'_content' => 'Can\'t connect to ___censored____.com:443

',
'_rc' => 500,
'_headers' => bless( {
'client-warning' => 'Internal response',
'client-date' => 'Tue, 10 Jun 2014 00:47:12 GMT',
'content-type' => 'text/plain',
'::std_case' => {
'client-warning' => 'Client-Warning',
'client-date' => 'Client-Date'
}
}, 'HTTP::Headers' ),
'_msg' => 'Can\'t connect to ___censored__:443',
'_request' => bless( {
'_content' => 'Test Machine - Test Company"Toast Test - Disregard this message"',
'_uri' => bless( do{\(my $o = '__censored :)__')}, 'URI::https' ),
'_headers' => bless( {
'user-agent' => 'libwww-perl/6.06',
'x-wns-cache-policy' => 'no-cache',
'x-wns-type' => 'wns/toast',
'x-wns-ttl' => 600,

I've tried setting environment variables SSL_VERIFY_NONE and PERL_LWP_SSL_VERIFY_HOSTNAME several different ways, no luck.

Any meditations from the monks?