Yes. The ability to move granted by rogue scout is the same as the general ability to move on your turn
Speed versus movement
Just to clear things up for the rest of this answer, Jeremy Crawford has pithily stated the difference between the two in this tweet:
Speed ≠ movement. Speed is a potential. Movement is an actuality.
Freedom of movement on your turn
Let us look at how using freedom of movement works on your turn normally.
The PHB states a rule that allows you to, on your turn, move up to your speed:
On your turn you can move a distance up to your speed...Your movement can include [...]
Thus, a rule has granted the character the ability to move on their turn only.
And, then you could obviously use 5 feet of that movement to use this part of freedom of movement if you so desired on your turn:
The target can also spend 5 feet of movement to automatically escape from nonmagical restraints, such as manacles or a creature that has it grappled
Freedom of movement with skirmisher reaction on your opponent's turn
We established above that the general rules establish that you can only move on your turn. However, there are features that override that general rule such as Skirmisher Reaction:
You can move up to half your speed as a reaction when an enemy ends its turn within 5 feet of you. This movement doesn't provoke opportunity attacks.
Note the similarities between this specific rule:
You can move up to half your speed as a reaction...
and the general one:
On your turn you can move a distance up to your speed...
Both these things are granting the exact same ability, the ability to move some distance, to the character, just under different conditions. There is no difference either specified or implied between the way in which either ability says "you can move" that would differentiate it from the other.
It makes no sense not to allow "spend movement" with skirmisher reaction
Even the language used supports the fact that you are able to "spend movement" with the movement granted to you by skirmisher reaction.
Freedom of movement says:
spend 5 feet of movement
The general rule on speed says:
And skirmisher reaction says:
If even the description says all of these are "movement", why can you not spend them the same way: as movement?
If you do not allow the character to "spend movement" with skirmisher reaction then they also would not be able to stand up from prone, move while prone, or move accross difficult terrain in any way. None of which makes sense, since, again, there is no essential difference between the movement granted on your turn and the movement granted by skirmisher reaction.
Thus, if you can spend 5ft of movement for freedom of movement on your turn, there is no reason that you cannot do it as a reaction at the end of your enemy's turn with skirmisher reaction.