| routers were not secure and that they had security issues|
Where on earth did you hear that. Even it were true, inserting a router will make you no less secure than what you have now.
The reality is that routers improves security because of the Network Address Translation mentioned earlier and most home routers have a firewall built in. The only insecure aspect of routers is if a wireless router is configured with an open Wi-fi connection. Something that is easily rectified with a reading of the manual. If you have no intention of connecting via wireless a simple wired router will do the job or a wireless router with wireless disabled.
On thing is certain that you cannot connect two device simultaneously to a single internet connection without a router unless you want to get into the complexities of Internet Connection Sharing via software.