Third Party Bluetooth Mice/Keyboards Not Working On Startup

I bought a Logitech mouse that's made for Mac, and I have no complaints. It works just fine if I am logged in. But when I restart my iMac I am not able to use the mouse/move the cursor on-screen unless I use an Apple-branded mouse or keyboard to click on my user and enter my password. Once I log in I am totally able to use third-party keyboards and mice just fine. Does anybody know why this is? I am thinking that perhaps it is a security setting that is not allowing any third-party Bluetooth device to connect until I log in. Thoughts? This is a huge annoyance.