From their FAQ.
Do I need a powerful PC to play these high definition games? CORE®ONLINE will work on most modern connected Windows PCs using an updated major browser. As the service is free, we recommend that you try it out on your system and see if you like it.
Is CORE®ONLINE compatible with Mac or Linux? At the moment only select titles are available to Mac and Linux users. If you would like to see more of our games on a particular platform then let us know!
Which browsers is CORE®ONLINE configured for? CORE®ONLINE is configured to run on most updated browsers using the Square Enix Secure Launcher plug-in (including Internet Explorer and Firefox) or on Google Chrome via Native Client without any installation at all.
What is "Square Enix Secure Launcher" and why do I need it to run the game? Square Enix Secure Launcher is a secure browser plug-in developed by Square Enix specifically to run CORE®ONLINE games, offering full native game performance in all mainstream web browsers. SESL is currently only available on Windows.