If you wish to open all the requests to LiveAgent always with https protocol then try adding the following lines into your .htaccess files in the LiveAgent root folder and in its /agent folder:
RewriteCond %{HTTP:X-Forwarded-Proto} !=https
RewriteRule (.*) https://%{SERVER_NAME}/$1 [redirect=permanent,last]
Add those lines right under the line with:
RewriteEngine on