I have to agree with Flowers that not caring in the sense of not hanging on to any opinions is the way to go.
Poppers said:
Being judgemental is not something that a person "is", but rather is a "point of view" that flows through the identity structure one has constructed for oneself .. It's the identity structure (what I call ego) that keeps people in separation from one another.
I would simply say it is identification with structure or form, which is a slightly different thought from identity structure - as the actual structure isn't as important as whether you identify with it. In this case thought forms that creates the phenomenon of ego, since it doesn't really exist as an entity - it's just this dynamic, an activity.
You have asked an important question - how to overcome it. And you have found the answer, though you may not realize it yet. The answer is basically very simple: just become alert to the arising of judgement of ANY kind.
But is judgement (or intolerance even) bad, something you need to get rid of? What is there to say it is something that needs to be overcome except another judgement? See it becomes very subtle doesn't it, it's tricky to frame this stuff with ideas. It should be pointed out that what judgement is is essentially unconscious, simplistic information processing. You've got your little standard of judgement, and then you compare something you see with it and basically you just have either approval (if it fits) or disapproval - of course meanwhile reality is as it is regardless of what you think of it. Funny it should be the other way around - ones standard of truth should really be based on that, why should reality conform to how you think? It may sound funny, but by 'not caring' I simply mean not taking it seriously. Even if it's not intended in this way, when a statement such as being aware of judgement or thoughts, breath or anything comes across as a technique, something to do, then you're stuck again. It's just one thought leading to another, even if it is a very subtle state of consciousness. Of course when you're there with someone in real time you can just ask a very simple and open question to actually inquire, that's kind of hard to do in a post since it would probably not last very long if anything. But remember that "don't know mind", it's not knowing that frees you from identification with thoughts, having different thoughts just makes you hooked to other, perhaps more sublime thoughts. For those who don't know what that means, it just means don't pretend you know when you don't.
There's a difference between just thinking a certain way and knowing.