Ok, I've done a quick pass through the code. It looks like the ECalerts_run.php needs a couple of changes to support the HTTPS only on weather.gc.ca, and bypass any curl difficulties with host/peer SSL validation (as not all hosters have that set correctly).
change:
$WWurl = 'http://weather.gc.ca/warnings/report_e.html?'; // EC URL for Warning & Watch & Statements data *** as of December 2013 ***
to:
$WWurl = 'https://weather.gc.ca/warnings/report_e.html?'; // EC URL for Warning & Watch & Statements data *** as of December 2013 ***
and change curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_URL, $URL);
to
curl_setopt($c, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($c, CURLOPT_URL, $URL);
curl_setopt($c, CURLOPT_SSL_VERIFYPEER,0);
curl_setopt($c, CURLOPT_SSL_VERIFYHOST,0);
curl_setopt($c, CURLOPT_USERAGENT, "Mozilla/5.0 (EC_Alerts V3.5)");
Give those a try...