If it's not possible, is there a way to configure the OpenWRT web server to allow or deny access to a page or directory depending the client ip address ?
typically on Apache-style webservers access control can be done with .htaccess. The webserver on Yun is the same as the one with OpenWrt. It is called uhttpd. Basic authentication is available via name and password.
Basic Authentication (httpd.conf)
If this does not serve your purpose, you can write a CGI to check your clients' IP. You can use Perl, Python, and Lua is the default for OpenWRT (and Yun). There are also various libraries available (such as IPsec - which allows and denies access on a per IP basis). And, of course, you can put your private data on a different port via uhttpd.
Given further thought you can also use:
NOTE: if the library list is useful, I would appreciate any feedback - especially any need improvements.