No. I don't believe you should, the reason: maintainable code.
I've seen people use properties within the defining class and at first all looks well. Then someone else comes along and adds features to the properties, then someone else comes along and tries to change the class, they don't fully understand the class and all hell breaks loose.
It shouldn't because maintenance teams should fully understand what they are trying to change but they are often looking at a different problem or error and the encapsulated property often escapes them. I've see this a lot and so never use properties internally.
They can also be a performance hog, what should be a simple lookup can turn nasty if someone puts database code in the properties - and I have seen people do that too!
The KISS principle is still valid after all these years...!