I note that there are two questions here. In the future you might consider posting two questions instead of combining two questions in one. When you combine questions like this often what happens is only the first one gets answered.
The first question is "why are private virtual methods illegal in C#?"
Here are the arguments against the "private virtual methods" feature:
1) private virtual is only useful when you have a nested derived class. This is a useful pattern, but far less common than the non-nested derived class situation.
2) If you desire to restrict the ability to override the method in non-nested derived classes then you can do so by restricting the ability of non-nested classes to derive from the base class; make all the base class constructors private. Therefore private virtual is not necessary to prevent overriding; protected virtual is sufficient, because the only derived classes will be nested.
3) If you desire to restrict the ability to call a method in a non-nested derived class then you can make the method internal virtual and then tell your coworkers to not use that method. It is irritating to have this not be enforced by the compiler, but the compiler does not enforce any other semantic constraint on how a method is supposed to be used either; getting the semantics right is your business, not the compiler's, and you have to enforce that with appropriate code reviews. Therefore private virtual is not necessary to prevent calling; internal virtual plus code reviews is sufficient.
4) It is possible to implement this pattern already with existing parts:
abstract class C
{
private int CF() { whatever; }
private Func<int> f;
public C() { f = CF; }
private int F() { return f(); }
private class D : C
{
private int DF() { whatever; }
public D() { f = DF; }
}
Now I have a method F which is effectively virtual, but can only be "overridden" by derived nested classes.
Since in every case either protected, internal or protected internal does the trick, private virtual is unnecessary. It's almost never the right thing to do, since you have to be already committed to using the nested derived class pattern. So, the language makes it illegal.
The arguments for are:
There have been times in real code when I've want a virtual method to be a private implementation detail of a class that I want to be extended both by non-nested internal classes and nested internal classes. Having to enforce the invariant that the internal method not be called by my coworkers is vexing; I'd like that to be enforced by the compiler without me having to jump through crazy hoops like making a field of delegate type, etc.
Also, there's simply the matter of consistency and orthogonality. It seems weird that two things that ought to be independent -- accessibility and virtualness -- have an effect on each other unnecessarily.
The arguments against the feature are pretty strong. The arguments for are pretty weak. Therefore, no such feature. I'd personally like it very much, but I totally understand why the design team has never taken me up on it. It's not worth the cost, and I would hate to not ship a better feature because we spent budget on a feature that benefits almost no one.
The second question is "Why in C# are you not able to override a private virtual method in a derived non-nested class?"
There are several reasons.
1) Because you can only override what you can see. A private method is a private implementation detail of a base class and must not be accessible.
2) Because allowing that has serious security implications. Remember, in C++ you almost always compile code into an application all at once. You have the source code for everything; everything is essentially "internal" from the C# perspective most of the time. In C#, that's not at all the case. Third party assemblies can easily get at public types from libraries and produce novel extensions to those classes which can then be used seamlessly in place of instances of the base class. Since virtual methods effectively change the behaviour of a class, any code which depends for security reasons on invariants of that class needs to be carefully designed so that they do not depend on invariants guaranteed by the base class. Restricting accessibility of virtual methods helps ensure that invariants of those methods are maintained.
3) Because allowing that provides another form of the brittle base class problem. C# has been carefully designed to be less susceptible to the brittle base class problem than other OO languages. If an inaccessible virtual method could be overridden in a derived class then that private implementation detail of the base class becomes a breaking change if altered. Providers of base classes should be free to change their internal details without worrying overmuch that they've broken derived classes which depend on them; ideally only the public, documented interface to a type needs to be maintained when implementation details change.