Breeding is sexual reproduction. Parthenogenic whiptail lizards lay eggs and reproduce themselves, but they don't breed.

Only our gametes can undergo meiosis and recombination, the necessary processes for breeding. Our other cells can engage in mitosis, duplication. There are only the two processes, which is why I say this is fundamentally incorrect.

The queen and drones are the gametes of hymenopterans in a very real sense. They have cells which can undergo meiosis and recombination, workers do not.

In fact, in some unusual species of ant, there are motile and fertile workers, in addition to and sometimes instead of the sessile queen. These are called... gamergates.

This is glossing over some of the weirdness found in various hymenopterans, it's an ancient and diverse clade.

Individual ants are animals, that's not controversial. Eusocial animals are less individual than any other animal, and their social groups are more like a single organism than the social groups of any other sort of animal. This, too, is not controversial.