Use a subdomain of your company's registered domain for internal machines whose names you do not want available on the Internet. (Then, of course, only host those names on your internal DNS servers.) Here are some examples for the fictitious Example Corporation.
I used "corp" to signify that this subdomain described machines on the internal corporate network, but you could use anything you want here, such as "internal": client1.internal.example.com.
Remember, too, that DNS zones and subdomains do not have to align with your network numbering scheme. My company, for example, has 37 locations, each with its own subnet, but all locations use the same (internal) domain name. Conversely, you could have only one or a few subnets, but many peer internal domains or levels of subdomains to help you organize your machines.