If I save a EPS vector file in Illustrator CC version will it open up in any Illustrator version (e.g. CS4, CS5, CS6)? Or do I need to save separate EPS files for each version of Illustrator?
In general... it may open, but will present a warning for legacy versions -- something along the lines of "This file was generated with a newer version of Illustrator. Some object may not appear correctly."
Then it often opens with objects completely missing or flattened and may even rasterize objects in some instances.
If you want a solid EPS to open in any version of Illustrator, think oldest version. Then save to that version EPS format. Anything newer will open the old format EPS fine. So if you want to ensure AICS4 compatibility, save to an AICS4 EPS format (there is that option when you save an EPS). Then anything newer than CS4 will open the EPS without an issue.
This often means you need to construct the EPS with the limitations of the old version in mind. For example, saving to Illustrator 8 EPS was standard for many, many years, and is still fairly common. But to properly save to AI8 EPS, you need to remove all transparency and blend modes from the artwork.
Also note that "CC" isn't necessarily enough of a discriminating factor. EPS files from CC2017 may still be problem if opened in CC2014.
Small Tip: If you are saving to a legacy version EPS, always retain the original .ai file and work on that, not the EPS. In addition, it's really wise to open the saved EPS and see how things are constructed. There are often some minor changes you can make in the original file which will make the legacy EPS either less complex or to improve overall construction within the EPS format - things like, manually flattening a specific part of artwork, or expanding a stroke.. minor things that are just specifically so the EPS is better constructed. I never save a legacy EPS without then opening it to see how the art has been reconstructed. I then go back to the original .ai file to make any changes and re-save a copy as an EPS. I don't make any changes in the EPS itself.