Is there a formal name for "I'm right, don't try to challenge me" / "if you push against me I will be 100x more stubborn" personality trait? Some persons gets really annoyed if we try to demonstrate our thesis to them or to put in light that they are behaving in a childish way. 

Dale Carnegie, in his most famous book: How to win friends and influence people, wrote that it's almost completely unuseful to try to criticize others even if we are completely right. Even if we are trying to help them. The psychological reason behind is that we hurt their self-esteem / ego.

But when does this come to be extreme? When people are extremely stubborn and any attempt to change their mind or to show a different point of view becomes like a fight with them? 
(Of course this definition is usually related to some of their belief on which they can become oversensitive).

It can happen when they have a strong need to confirm their beliefs, but also if their belief is that to insist is disrespectful. 

I think this trait is very frequent in Narcyssistic (which thinks to be super rational).

This trait can be explicitated both in an aggressive way and in a passive resistence. The description provided by wikipedia:


  In psychology, passive-aggressive behavior is characterized by a
  habitual pattern of passive resistance to expected work requirements,
  opposition, stubbornness, and negativistic attitudes in response to
  requirements for normal performance levels expected of others. Most
  frequently it occurs in the workplace where resistance is exhibited by
  such indirect behaviors as procrastination, forgetfulness, and
  purposeful inefficiency, especially in reaction to demands by
  authority figures, but it can also occur in interpersonal contexts.
Defensive Pride I would say match your question title.