I found the sentence whose Greek was referenced in comments in the OP.
In Regimen, XXIV,
> The soul of man, as I have already said, being a blend of fire and water, and the parts of man, enter into every animal that breathes, and in particular into every man, whether young or old. But it does not grow equally in all;
Wikipedia lists this book as being "Works of the age or spirit of Hippocrates". Note that,
> The Hippocratic Corpus (Latin: Corpus Hippocraticum), or Hippocratic Collection, is a collection of around 60 early Ancient Greek medical works strongly associated with the physician Hippocrates and his teachings. They are, however, varied in content, age and style, and are largely of unknown authorship.
In summary,
* Hippocrates, or someone very like him and at about the same time, did say that
* The English translation cited in the OP is concise (abbreviated) and quite loose.