No.
YourSiteOurServer.com does not change your domain nameservers for you when you sign up for a new account. You must change your nameservers for your domain to point to YourSiteOurServer.com manually. You can change the DNS settings for your domain at your domain registrar (the company through which you purchased your domain), changing them to ours: ns1.yoursiteourserver.com and ns2.yoursiteourserver.com.
If you have registered a new domain with YourSiteOurServer.com, the DNS entries will already be set. If you transferred your registration to YourSiteOurServer.com, you can update the DNS entries in the Customer Support Portal when you are ready to switch your site over to your new account.