What's the word for this?
People in the same nation as me.

I met my fellow countrymen when I traveled to Fiji.

You are my fellow countryman.

Is men neutral for women too? Is there a better word?

Question

What's the word for this?
People in the same nation as me.

I met my fellow countrymen when I traveled to Fiji.

You are my fellow countryman.

Is men neutral for women too? Is there a better word?

Answer

"compatriot" is one word that fits the definition very well.  It is a good word to use in an essay but could be a bit too formal in a casual conversation.

Answer

Hi! In this specific case  - yes, you can apply it to both men and women. However, if you want to sound completely gender neutral you can use "fellow citizen" instead. I hope that was helpful! Please feel free to get in touch with me if you will have more questions.

Also, here's a good website where you can look up synonyms: 
https://www.thesaurus.com/