Which word came first, cockroach or la cucaracha?

No henroaches, but you're not far off the mark with the idea. "Cucaracha" came first. First known use of "cockroach" (or the word that would become it, anyway) is 1624 -- colonist John Smith, writing about the bug, said it was "called by the Spaniards a Cacarootch."

That was probably what's called folk etymology (basically, borrowing a foreign word, but changing its spelling and/or pronunciation, either by mistake, or because the original is difficult to pronounce).

Anyway, 1657 was the first time it was spelled "cockroche," with the first part simply meaning "rooster," and "roche" being an old spelling of "roach" -- which, back then, referred to a type of fish. Why these words were chosen isn't clear, other than maybe someone decided to replace "cacarootch" with two similar-sounding English words.