I'm going to get a lot of heat for this comment and perhaps others will want me to explain further in detail (which I would rather not).......
YES.
Any shared environment adds to the risk to every site on the server. This includes shared web host, but also VPS host as well. Does not matter how good you've configured the setup... The risk is there. You can limit the risk, but never to a complete zero (never as good as a dedicated).
Their statement is generally correct, except Apache isn't the root of all evil. They are forgetting MySQL, PHP, WHMCS, Xen, OpenVZ, Cpanel with WHM, and there are other routes one could use to compromise a shared environment.
A complete dedicated environment has the least risk, when properly secured.