I was having problems installing atom packages at work (on a Windows 7 machine) and I really wanted to have syntax highlighting in my React project. Turns out it's really simple to manually install a package.
Download the package you want. It's probably in GitHub. I wanted the Atom React Package.
Extract the zip file and put it in the packages folder. You can find this folder from the install packages screen:
Open a command window in the folder you just put in the packages directory.
Run apm link
Restart Atom and you're done.