Extra Attack only modifies a regular Attack action
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
I think the fact that "Attack" is capitalized seems to implicate the Attack action under actions in combat vs. any attack.
Horde Breaker could be used in conjunction with Whirlwind Attack
Once on each of your turns when you make a weapon attack, you can make another attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.
Since Horde Breaker triggers off a "weapon attack" not an "Attack action" you should be able to use Horde Breaker to target one of the enemies next to you. Its simply triggering off one of the attacks made as part of the whirlwind.
Optimization breakdown
Since all Hunters get either Volley or Whirlwind, the Extra Attack class feature is there as a stopgap in the meantime and also a fallback for when there is only one enemy in range to be targeted. Horde Breaker should still be viable even though its utility will diminish in comparison upon reaching level 11 as a ranger, but even then its still one more attack per turn.
I run a group with several melee combatants, including a melee ranger. We have not gotten to L11, but I can tell you from my experience that an ability that would allow you to attack more than 2 enemies is incredibly useful.
Here's the thing, as you level up, you're supposed to not only face greater threat level foes, but you're going to be facing more low level foes at once. Adding powers to a power list that allow a ranger to effectively operate as crowd control, especially when they are dealing more damage and can knock a low level foe out completely, is a great power.
Think about this for a moment, let's take the lowly goblin a CR 1/4 foe. Using a party of 4 characters, at L1, 4 goblins are a deadly encounter (50xp *2). At L5 it takes 27 of them to be a deadly foe, and by L11 you need 72 of them to be a deadly encounter.
While no, you're not likely to face off 72 goblins at L11, you may face off against 10 CR 2 opponents in a deadly encounter. This means there is likely to be a situation where you will have 3+ foes around you if you are engaging in melee combat.
From experience, I can tell you that this is a frequent occurrence for melee characters, they are often swarmed by 3, 4 or even more low level foes and potentially need to be able to deal damage to all of them, especially if they are much lower level and can be killed in one shot.
As far as whether it's a mechanically sound choice over Volley, that likely depends on both your character and your campaign. If your character is never going to use a ranged AOE (his ranged attack isn't good or he is a front line fighter), that's not going to be a good choice for him. However, if your character is a ranged ranger, Volley is a no brainer over this.
Best Answer
Sage Advice sets the precedent that Volley is one attack with multiple rolls
At level 11, a Hunter ranger gets a "Multiattack" feature per the PHB. That multiattack manifests in one of two ways: Either a ranged type (Volley) or a melee type (Whirlwind Attack):
In essence, a Hunter ranger gets the Multiattack ability but can tailor it to match the melee or ranged combat style they have chosen to develop. The wording of the rules are identical other than the ranged/melee qualifications.
Both are unclear as to whether the ranged/melee "attack against any number of creatures" is a singular attack (which targets multiple creatures), or multiple attacks against each of the creatures.
The Sage Advice Compendium specifically allows only the former interpretation for the Whirlwind Attack feature:
Because there is common wording between the two variations of the Multiattack feature and because the Sage Advice Compendium informs us that one of these two variants is one attack with multiple rolls, the other (Volley) must necessarily be treated the same way.
Lead rules designer Jeremy Crawford tweeted in September 2014 that this interpretation matches designer intent:
(Hat tip Speedkat & Rubiksmoose.)