Private properties can be very useful to encapsulate the behaviour of things which are internal to your class. Just because they are private doesn't mean that you shouldn't take advantage of the syntactic sugar that properties give you.
The example given is a poor one, however. Boilerplate property getters and setters like this are almost always a bad idea. If you are using C# 3.0 or later then, auto properties are a much better idea:
private string Whatever { get; set; };
This is shorter, cleaner, and much more readable. In fact it is barely longer than just declaring the backing variable.
Most importantly though, auto properties can be converted to full properties at any time without changing the semantics of the rest of the program! Thus if you need to add validation or error handling, you can do so easily.
As it is, I always thought it was a shame that you couldn't have a read only property, to enforce only being able to write to the value in the constructor, (I prefer immutable types) but this was added in C# 6.0 as "Auto-Property Initializers", which allows you to do:
private string Whatever { get; } = ...;
or
private string Whatever { get; };
along with
Whatever = ...;
in the constructor.