Yes, that is the neutral variant. "Matěj hlad nemá." stresses the word "nemá". For example, it is a response to someone who thought Matěj is hungry. Or, another example, when there is a group of people including Matěj, most of them are hungry, and you say that Matěj is not hungry.
I've never noticed differences in word order between Czech and Slovak. There are differences in everything else, but not word order.
However, what do you mean here? The sentence is literally "Matěj hlad nemá" - same as what your partner said.
The default word order is, of course, "Matěj nemá hlad". The word order in this exercise with "nemá" at the end is used when you want to stress that he is not hungry, while somebody else is or while somebody claims that Matěj is hungry.
Sorry, i meant to write that she would only say Matěj nemá hlad. She recognises that the order can be changed but she would never consider saying Matěj hlad nemá. That just made me curious to know if i should avoid taking advice from a Slovak when learning Czech despite her being able to speak Czech fluently.
I don't know. It might be because Slovak prefers to say "on nie je hladný" (literally he isn't hungry) instead of "on nemá hlad". Or perhaps your partner just doesn't imagine the proper context where the reversed order is applicable.
Definitely don't consult conjunctions with her, such as "když", "jestli", "pokud", "až" - Slovaks very often use them wrong when they speak Czech ;)