In my desktop application I added access to various internet resources using boost::asio. All i do is sending http requests (i.e to map tile servers) and read the results. M
It seems that sample is merely a show-off of what Boost ASIO can be used for but is likely not intended to be used as-is. You should probably use a complete library that handles not only HTTP proxies, but also HTTP redirects, compression, and so on.
HTTP is a complex thing: without doing so, chances are high that you will get news from another client soon with another problem.
I found cppnetlib which looks promising and is based on Boost ASIO not sure it handles proxies though.
There is also libcurl but I don't know if it can easily be integrated with Boost ASIO.