Hi there,
In chatbox/include/structure.php, change lines 38 & 39 to read:
<a href="#!/lively-chat-support" target="livelychatsupport-iframe" data-href="<?php echo preg_replace('#^https?:#', '', plugins_url("lively-chat-support/chatbox/convos.php?open=")); ?>" class="livelychatsupport-close">—</a>
<a href="#!/lively-chat-support" target="livelychatsupport-iframe" data-href="<?php echo preg_replace('#^https?:#', '', plugins_url("lively-chat-support/chatbox/convos.php?open=")); ?>" class="livelychatsupport-open">
I’ve fixed in for the next release… but version 2.0 is coming out soon, which is a complete rewrite (and will work over HTTPS)!