Boost not updating without www site after clearing cache
You need to clear the Boost cached pages, i.e. the static pages. You can do that on the Boost settings page at the bottom under Clear Boost's Database &amp; File Cache. If you only have changed one page you can just clear this page by enabling the boost block Boost: Pages cache configuration.

Edit: After a bit more thinking I am not sure clearing the Boost cache will fix your issue. However, do yourself a favor and change your .htaccess rules to redirect users to either with www or without www. I would suggest with as that is working for you. Having both counts as duplicated content in search engines and you risk to loose ranking.

To fix this, uncomment the following part in your .htaccess file (around line 81), or add it if it's missing:

RewriteCond %{HTTP_HOST} !^www\. [NC]
RewriteRule ^ http://www.%{HTTP_HOST}%{REQUEST_URI} [L,R=301]