Do I need to replace this toilet ring?
Yes - the rectangular hole on the right leads to a curved slot that holds the bolts used to secure your toilet.  Your slot is broken which means the right side of the toilet will not be secure.  This could (read will) lead to a rocking toilet (not in a good way) which will be followed by leaks (um, eww) and rotted subfloors or worse.  

A repair flange like this will work, but a full ring like this is better.

And I'd lean toward a stainless repair kit so there is less worry in the future.