That is completely different than what break away is.
Like I said above, I think you might want to review the section on the different types of actions in the rulebook (page 9).
There are five types of actions: close combat, ranged combat, power, move, and free.
You give actions to your characters to do stuff. The stuff the action allow you to do can vary, but the use of the general term action simply makes it so that all types are covered.
The rulebook says (page 8):
"If a character occupying a square adjacent to one or more opposing characters is given an action and attempts to move, that character must successfully break away before it can move, as shown in Figure 8."
That means it doesn't matter what type of action was given. You could have been given any type of action. Regardless of the type of action you are given, if it involves movement, then if you are adjacent to an opponent you must break away before you can move.
The key thing you just said is "in my mind" which is exactly what I was mentioning before. You are trying to visualize what you think the power is representing rather than just taking it at face value. You are doing no movement within that free action, so there is no break away applicable.

No, I know the types of actions. And I stated the same thing for break away you did. It says if they are given an action AND attempt to move. So, that, to me, is two different things. If it said "when a character is given an action to attempt a movement", I would have not have posted the question. But since it says "given an action and attempts to move" is my deal. In this case, the free action during hypo. I read it as two unrelated things. Not a power action or free action that involves movement or a move action. Just any action, then a movement afterwards.

Edit: I am not including the whole description bc we all know break away is normally when adjacent to one or more opposing characters. I am just focusing on specific wordings in it.

There is one more key element that you are neglecting.
Break away is triggered before movement begins.
There is no movement beginning after the resolution of the free action. It is still the same movement that you started right after you gave the character its power action.
You only have one movement during HSS.

Thanks for being patient Harpua. Ok. Where does it says break away happens before movement begins? Because I just read the "...given an action and attempts to move..." part. It doesn't state it happens at the beginning of movement. Just an attempt to move.

Maybe if I describe it in a different way.

Imagine that your HSSer begins the action next to an opponent, specifically, the opponent he wants to attack with the HSS.

How it works:
1) Give the HSSer the power action to activate HSS.
2) Roll break away (success)
3) Begin movement
4) Move 0 squares
5) Give free action to attack
6) Attack
7) Resolve the free action
8) Move away
9) End movement (began at #3)
10) Resolve power action

How it does not work
1) Give the HSSer the power action to activate HSS.
2) Give free action to attack
3) Attack
4) Resolve the free action
5) Roll break away (success)
6) Begin movement
7) Move away
8) End movement (began at #3)
9) Resolve power action

The attack is not an action.... It never stayed out is an action it is simply an attack. They're is a difference

See the quote from the rulebook above:
"If a character occupying a square adjacent to one or more opposing characters is given an action and attempts to move, that character must successfully break away before it can move, as shown in Figure 8."

The attack is a free action.
"During this move, this character may, as a free action, make one close combat or ranged combat attack with its range value halved for the attack."

I understand how hypo works when starting adjacent. I understand the free action happens during movement. And I understand it says before it can move. But is is referring to the attempt to move earlier stated in the rule. The piece that hypos in is not starting another movement. I understand that. It's all one movement. But, he is moving from one square to another, and must move away during hypo. As you stated in your explanation. He is still moving. And breakaway is as you stated. After moving up with hypo, He is occupying a square adjacent to an opposing character, is given an action (our free action close combat attack) AND is attempting to move. moving away to continue his movement in hypo. It doesn't say attempts to make a movement, it just says attempts to move. Which is what he does during hypo. But before THAT move, I feel he needs to roll break away. Because of a free action given while adjacent, and trying to move (not start another movement) with remaining speed value there after. I don't think I am making my point clear. I'm sorry. I am a more gifted speaker. But I understand all the rules. Just focusing on this wording.

HSS says that you use Improved Movement OO for that action.
"May move through squares adjacent to opposing characters, but still needs to break away normally."
Since your movement never stops you are just moving the squares adjacent to your opponent.

Cool. Got it. I don't know why it wasn't clicking. One more question. Plasticity shows up after actions resolve. Flash can continue to hypo away normally. But, if for some reason, he moves adjacent to that character with plasticity now showing, he stops. Correct? So he is adjacent, but plasticity says you need to enter a square adjacent. Since he is already there, doesn't matter. But, if he moves to another square that is also adjacent, stop movement.

Perhaps thinking of it this way will help:

After a character moves a single square, does he need to roll break away in the second square (if still adjacent to that character)? Or if he moved 7 squares, does he need to roll break away to get to the 8th square, if he is Hypersonic'ing by some character? If you can accept that a character can Hypersonic move, needing to break away just once, if he were simply moving - then can you accept that he can do SOMETHING ELSE TOO, in addition to moving, when he occupies a selected square?

If you can allow a character to make a single break away roll, even if they attempt to "step behind" an opposing character, ie moving into contiguous adjacent squares, -- then allow a figure to do something else while they pass by too, like punch him in the face.

That is correct.