According to Merriam-Webster, both "jeez" and "geez" are correct, but in my opinion, Jeez is more correct since the term is a euphemism derived from the expression "Jesus" which some people use as an expression of shock or surprise.
I just wanna throw this thought out there. What is the definition of a "native English speaker?". When I think about it, its seems like a flawed definition. I say that because English is the universal language and you have perfectly fluent English speakers from just about every country on Earth. Would you consider a boy from China for example, who grew up in an English speaking house and with perfect English speaking abilities to be a native English speaker?