Yes

Horde Breaker states (PHB, p. 93, bold added):

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.

For both Green Flame Blade and Booming Blade, you do make a weapon attack:

You brandish the weapon used in the spell’s casting and make a melee attack with it

Note that you won't get any of the benefits of either spell a second time for the second attack, as explained in V2Blast's answer to this Q&A: Can Green-Flame Blade be cast twice with the Hunter ranger's Horde Breaker ability?

A concentration check is needed for each hit

A multiattack is separate attacks, so you'd have to roll for each attack that hits. The reason the PHB uses "sources of damage" instead of "each time hit" is because some damage sources use a saving throw instead of an attack roll, or automatically cause damage. So you need to make a concentration check, for each instance of damage taken.

Examples

• Scorching Ray - You create three rays of fire and hurl them at targets within range. You can hurl them at one target or several. Make a ranged spell attack for each ray. On a hit, the target takes 2d6 fire damage. Each ray is a separate attack, and requires a separate concentration check.

• Sword of Lightning - 1d8 slashing + 1d4 lightning damage. Two damage types, but one source (the sword attack); one concentration check.

• Dragon - Wing Attack [...] Each creature within 15 feet of the dragon must succeed on a DC 23 Dexterity saving throw or take 15 (2d6 + 8) bludgeoning damage and be knocked prone. One source of damage. One concentration check on a failed save.

Edge Case

• Imp - Sting [...] Hit: 5 (1d4 + 3) piercing damage, and the target must make on a DC 11 Constitution saving throw, taking 10 (3d6) poison damage on a failed save, or half as much damage on a successful one. By RAW, one hit and one save, thus two concentration checks; however, it can be easily argued for a house ruling that it is one damage source since the save only determines the amount of poison damage not whether there is poison damage.

• Magic Missile - Multiple darts that hit simultaneously. Since the darts automatically hit, there is no attack roll nor saving throw. Thus, ambiguous as to whether it is one damage source or multiple. There are valid arguments for ruling both ways. However, Jeremy Crawford rules that each dart requires a separate concentration check. As a lead designer, his tweets can be considered RAW.