Apparently the author who put forth the Zuni-Japanese hypothesis is not a linguist, so... yeah.
Linguistics is one of those areas where the layperson thinks they know a lot more about it than they actually do (after all, they can speak English and they even learned a foreign language!). If someone's putting forth a theory like that, I'm skeptical of it if they don't have much background because the methods that seem to make sense at first to use for historical linguistics don't actually hold up very well. Like picking out random words and finding similarities - if two languages have reasonably similar phonologies (and in this case, they both share the five vowel system [a e i o u], which is the most common vowel system) you'll find at least some words that match, or come close to matching.
What would be more convincing is if, for example, she could find some kind of consistent sound correspondence between the two languages. Looking at English and Swedish, you can find a strong correspondence between words spelled with sh in English and words spelled with sk in Swedish (if followed by a front vowel [i e y ö j] it's actually pronounced more like sh):
English | Swedish |
shall | ska/skall |
shoot | skjuta |
shit | skit |
shoe | sko |
-ship (suffix) | -skip |
sharp | skarp |
Or when you look at Germanic languages and Romance languages, there is a correspondence between [f] and [p], th and [t], and [h] and [k]:
Germanic | Latin/Romance |
father | pater |
mother | mater |
heart | cor |
hound | canis |
thou | tu |
foot | ped |
head <(Old English) heafod | caput |
horse | currus (chariot) |
throw | terere |
You can find patterns like that between most languages that are related - like Mandarin and Cantonese (Mandarin only allows nasal consonants at the end of syllables, while Cantonese allows a number of consonants - but one of the tones of Mandarin corresponds to syllables that end in consonants in Cantonese).
Also, given that Japanese has a fairly long written history, she would be able to look at Middle Japanese, which would be from the time period she was suggesting the Japanese settlers came to the Americas, and it's fairly well-attested, so there would be little reason to use modern Japanese words for her comparisons (sometimes words can sound superficially similar in the modern languages but are found to have very different origins).
One example where you might assume there's a relationship between admire and admiral - but admiral actually comes from Arabic
amir, emir (like in the United Arab Emirates) and picked up a d due to spurious association with Latin).