What are false unchanging entities?

The canonical example is the Soul or atman of persons that is believed by many religious people to be an unchanging entity that exists on its own.

Another example from many modern scientists are the sub-atomic particles (such as quarks or electrons) that appear to be unchanging entities that exist on their own.

Another example I heard on this site is the number 2. Someone was arguing that the number 2 is an unchanging entity that exists on its own. For that matter, I think they were saying the same of all numbers or at least the positive integers.

Actually, this is the default - if subconscious - belief of nearly everyone for just about everything. Things appear to us as solid and real and existing independently where in fact in reality they are not solid or real nor do they exist independently. My favorite non-Buddhist example of this is the Ship of Theseus. Everything that exists does so in the same way as Theseus' ship without the slightest bit of an unchanging essence behind it. And yet, by default we think that existing things do exist via self-sustaining essences. This is our ignorance.

If you really think about it, we exist just like Theseus' ship. When we are born we are given a name which generally sticks with us all the way through childhood and adulthood and until we die. Over the coarse of our lives every single atom in our bodies is replaced many, many, many times over. Our bodies change until they are almost unrecognizable. Our minds change and our attitudes and beliefs change. There is nothing about us that isn't constantly in a state of flux throughout our lives. Yet, we hold onto this view that we are the same person as we were when we were just a baby. That there is some fundamental aspect about ourselves that is unchanging and exists on its own. Often times when nostalgia hits us we have this naive wish thinking, "If only I could go back and do this or do that again" when the "I" that is now is totally different than the "I" that was then. Still, we think this "I" hasn't changed and so when this "I" occurs to us it gives this definite sense that it is unchanging and permanent and exists separately from all the contingent factors that it utterly depends upon.

This is not the case. This is the false view of the self that we misconceive due to our ignorance.

The page you quoted from talks about mentions "the assembly of the five aggregates".

It's standard doctrine that there's no such thing as a permanent (not impermanent), independent (not "dependently originated") self: that's the so-called anatta doctrine, see for example What is the precise meaning of anatta?

The "five aggregates" are things like "form" (perhaps "body"), "feeling", "consciousness", etc. -- which, apparently, people mistake to be themselves -- e.g. "I am this body" (or "I have this body"), "I am (or I have) this consciousness", and so on. Buddhism teaches that these things (these aggregates, skandhas) aren't permanent; that taking them to be "self" is a cause of suffering, and not a good theory (or "right view") to hold or attach to.

Given that Thich Nhat Hanh is Vietnamese/Mahayana, and talking on the same page about Bodhisattvas and peace workers, I think that what he's aiming towards is a view of selflessness (i.e. not selfishness) and interdependence (i.e. not independence). I think that's obvious when you turn the page and read "We have to strip away all the barriers" and so on.

I think that both Thich Nhat Hanh, and His Holiness the Dalai Lama, write "popular" books intended to benefit a Western audience.

Anyway, to answer your question, I think the only false view of of "an unchanging entity that exists on its own", that he's talking about here, is the "self" (or "a false view of self"). For example this (my invention, this isn't quoting anyone else) might be one of these false views:

I'm me; and I'll remain 'me' until I die. And I remain me regardless of what other people do or even regardless of what I do.

I think that Thich Nhat Hanh's view is that we (people) are highly interdependent; we exist together, we affect each other; everything I have (a name, a body, an education) comes from other people. In practice there's little or no sense in which anyone is independent, and to imagine we are "sealed off like that not only isn't living, it isn't possible".

I suppose an example of how people aren't "independent" comes from this (non-Buddhist) doctrine, which was once semi-famous as a doctrine for Early Childhood Educators:

CHILDREN LEARN WHAT THEY LIVE by Dorothy Law Nolte

If a child lives with criticism,
he learns to condemn.

If a child lives with hostility,
he learns to fight.

[etc.]...

If a child lives with encouragement,
he learns to be confident.

If a child lives with tolerance,
he learns to be patient.

[etc.]...

If you live with serenity,
your child will live with peace of mind.

With what is your child living?