Horde Breaker can work with green-flame blade, but not in the way you think it does

When you make a weapon attack, Horde Breaker lets you make another attack with the same weapon against a different target 5 feet from the first one, as stated in the quote. It does not let you cast a spell - all it lets you do is make an additional weapon attack.

Green-flame blade, as mentioned in the quote, involves a melee attack with a weapon as part of it, with additional damage leaping to another target 5 feet from the one you hit with the attack.

However, let's say your multiclassed ranger/sorcerer character casts green-flame blade on target A (who has another enemy of yours, target B, next to them and also within reach of your weapon). Then as part of the cantrip, your character will make a weapon attack with a melee weapon against target A. This meets the requirement for Horde Breaker:

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.

Thus, you can make a second attack with the same weapon against target B.

In addition, as a result of the green-flame blade cantrip, if you hit target A with the first attack you make as part of the cantrip, then target B will automatically take damage from the green fire leaping to them.

If both attacks hit, then a 4th-level character (the lowest level at which you could do this, as a multiclassed Hunter ranger 3/sorcerer 1) would do the weapon attack's normal damage to target A, and target B would take the weapon attack's normal damage as well as fire damage equal to your spellcasting ability modifier. (One level later, at character level 5, both targets would take an additional 1d8 fire damage.)

In short: Horde Breaker lets you make one extra weapon attack against a target next to the first - nothing more. The weapon attack made as part of green-flame blade can meet the requirement to let you make this second weapon attack, but you can only cast the spell once with your action.

Flame Blade cannot be used with Booming Blade

Although diegetically the Flame Blade spell gives you a flaming weapon, you cannot use it for ordinary weapon attacks like what Booming Blade requires. The Flame Blade spell only gives you the ability to make Melee Spell Attacks with your action, dealing the amount of damage specified by the spell.

You evoke a fiery blade in your free hand. The blade is similar in size and shape to a scimitar, and it lasts for the duration. If you let go of the blade, it disappears, but you can evoke the blade again as a bonus action.

You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage.

—Flame Blade, Player's Handbook, pg. 242

This is in contrast to a spell like Shadow Blade, which does yield a weapon that can then be used with Booming Blade or Green Flame Blade, because the weapon provided by that spell is, for all mechanical purposes, a proper weapon.

You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60).

—Shadow Blade, Xanathar's Guide to Everything, pg. 164

"But what if my DM is super nice to me and lets me do this?"

If it were possible to use Flame Blade with Booming Blade, as a consequence of DM houserules, you would make a Melee Spell Attack using the Spellcasting ability associated with Flame Blade, which as a Druid spell, is Wisdom, and then add the Booming Blade effects on top of the Flame Blade damage. This is because Booming Blade calls for the user to make a normal melee attack with their held weapon, and uses all the normal rules for an attack made with that weapon. Since Flame Blade's action uses a Melee Spell Attack, you'd have to translate that to its use with Booming Blade.

Conversely, if you were to use a spell like Green Flame Blade, obtained as a Wizard spell, with your Flame Blade, you'd have the strange (but mechanically solvent) situation where Wisdom is used as the Ability modifier for the Attack Roll, and then Intelligence is used as the Damage Bonus on the secondary effect of Green Flame Blade.