We've been following the development of a third party Twitch client the past few months and it recently became available for beta testing. That app is called Blink and is still currently in beta and being worked on. However, another client has already made its way in BlackBerry World called eSporter, pipping Blink to the post. There isn't much to the description of eSporter but by the loosk of it you can login to your Twitch account and browse through your favorites, as well as by game or channels. You can also perform a search.
You can watch the video directly from within the app but there is also an option to view it in the browser if you wanted. To login, go to the Following tab found on along the bottom. eSporter is available for all BlackBerry 10 smartphones, however, on the BlackBerry Q10, Q5 and the Passport, the video displays in the top left and you can't seem to zoom in when viewing via the app. You're better off using the shortcut to view it in the browser as this gives you options to use "full" screen. I put in quotes as you know that you get the black areas on the top of the bottom due to the square resolution. If you have a BlackBerry Z30, Z10 or a Z3, you can rotate your device into landscape mode for better viewing.
eSporter is free to download and as I've said, it is available for all BlackBerry 10 devices.