A man should say "Jestem Gwatemalczykiem" (Nom: Gwatemalczyk), a woman should say "Jestem Gwatemalką" (Nom: Gwatemalka).
As we rarely encounter people from Guatemala in Poland, somehow a safer version (in my opinion) would be to say "Jestem/Pochodzę z Gwatemali" (I am/I come from Guatemala). With countries about which we don't talk much, Polish people may not be sure how to create a demonym (a noun for a person from that country).
Q: Jestem Amerykanką, a ty? - I am American/I am an American, and you?
A: Jestem Włochem/Włoszką - I am Italian/I am an Italian
A: Jestem Polką/Brytyjką/Angielką - I am Polish/I am British/I am English
Q: Jestem amerykanką, a ty?
A: Jestem sofą/kanapą/kanapką - I am a sofa/a couch/I am a sandwich :)
"I am American" works, we just use "an American" because it's correct and also it's visible then why we want a noun in Polish.
So, "New Zealand" = "Nowa Zelandia", with a male denonym "Nowozelandczyk" and a female "Nowozelandka". So for you it's "Jestem Nowozelandczykiem".
A safe way can always be to say "[Jestem/Pochodzę] z Nowej Zelandii".
"I am American, and you?" should be OK when speaking/writing English IMHO. As I noted above, it is more natural for me to say "I am English" than "I am an Englishman". That is a difference from Polish.
Being a New Zealander is different again; I don't think you could say "I am New Zealanderish", except perhaps as a joke.
It would be inappropriate, impolite and... quite disrespectful.
Jestem Amerykanką/Amerykaninem, a pan/pani/państwo? -
I am American, and you, Sir/Ma'am/Ladies and Gentlemen?
Jestem Amerykaninem, a pan, (panie) doktorze?/pani, (pani)
doktor? - I am an American, and you, doctor/you sir/ma'am?
Pan/Pani/Państwo are capitalized in personal notes or letters