As far as I can tell no one has asked this question before, so forgive me if they have.
This also assumes the character here does not have an extra attack, so they are probably level 2 or 3 in exclusively fighter. I do not think this would be a good use of action surge, but it is interesting nonetheless. I think this question boils down to if preparing a reaction attack is counted as you attacking on your turn and just delaying it or not.
There is no such thing as "reaction attack", so I'm going to assume you are talking about the "Ready" action, which allows you to "hold" an attack until a certain trigger happens, at which point you can use your reaction to "release" the attack.
The rules for Action Surge state :
On your turn, you can take one additional action on top of your regular action and a possible bonus action.
Unlike the action granted by Haste, there is no limitations on what kind of action you can use this additional action for. As such, you are free to use your surge action on the Ready action, in order to prepare an attack for outside your turn.
You don't need to attack to use Action Surge. This is different than features like crossbow expert which necessitate the Attack action:
When you use the Attack action and attack with a one handed weapon, you can use a bonus action to attack with a hand crossbow you are holding.
Since Action Surge doesn't have this requirement, you could certainly use the Ready action with it:
Sometimes you want to get the jump on a foe or wait for a particular circumstance before you act. To do so, you can take the Ready action on your turn, which lets you act using your reaction before the start of your next turn.
But you might not need to.
Based on the fact that you mentioned the Crossbow Expert feat, and the "counted as you attacking on your turn" bit, I suspect you are concerned about the Loading property of Crossbows. Loading is sometimes simplified to "You can only use it once per turn", but the actual text listed under Weapon Properties is as follows:
Loading. Because of the time required to load this weapon, you can fire only one piece of ammunition from it when you use an action, bonus action, or reaction to fire it, regardless of the number of attacks you can normally make.
In other words, if you use your action to Attack with a crossbow, you can Action Surge and Attack again with the same crossbow because you would still only fire the crossbow once per action. Using the Action Surge to Ready an attack also works, but it isn't necessary if you just want to attack with the crossbow twice.
