My clients still use Photoshop CS6, if I register to have Photoshop CC will my clients be able to open and work on my files?
If you are providing a standard format, such as JPEG, PNG, etc., they will have no problem.
If you are sending them PSD or multi-layered TIFF files, for almost everything you put into a document, Photoshop CS6 will be able to open the PSD and work without problem. However there are a few things, such as setting up a layer stroke or advanced uses of Smart Objects that are new in Photoshop CC, that the older version will not be able to understand.
As a general guideline, if you do not do things in your documents with new features only available in Photoshop CC that you were not doing in Photoshop CS6, you will have no problems with backwards compatibility.
Something you can do to check, yourself: Install Photoshop CS6 AND Photoshop CC. They can both co-exist on your system without problems, and you can run one or the other. Then you can try opening a file in Photoshop CS6 that you have saved from Photoshop CC yourself if you have doubts.