Although "Eyvallah" is not actually a Turkish word, it's origin is from Arabic, it is mostly used by males to say "Okay". Women can also use it, it won't sound strange but they prefer saying "Tamam" instead. Turkish is a gender-neutral language. You won't see any Masculine or Feminine ways of speech in this language. There are male and female nouns for animals "like inek/dana" but that's not about gender-specific speech.

yes, generally men use this word (eyvallah), women use so so rarely. we dont have any word that only men can use. every word women can use also.

Salam, it's from the Arabic إى والله