Did you find the way to output the captcha over https?
same situation.
Nope, I never found a way around it. I had to create another side bar widget with a non-captcha contact form so the pages would show as secure.
Here is a method that worked for me. See the following post for the source of this solution.
http://ww.wp.xz.cn/support/topic/wp_content_url-based-defines-do-not-work-over-ssl-https
Essentially, I edited wp-includes/default-constants.php and changed the wp_plugin_directory_constants function as follows:
function wp_plugin_directory_constants( ) {
/**
if ( !defined('WP_CONTENT_URL') )
define( 'WP_CONTENT_URL', get_option('siteurl') . '/wp-content'); // full url - WP_CONTENT_DIR is defined further up
**/
if ( !defined('WP_CONTENT_URL') )
define( 'WP_CONTENT_URL', site_url( 'wp-content') ); // full url - WP_CONTENT_DIR is define with https or http per connection
Hopefully this will be helpful to others.