When everyone is out in a batsu the annoucer says something that sounds like "zen outto". I’m wondering what exactly is being said and does the "zen" correlate to zen buddism?
It’s saying "zenin" which means "all members" or "everyone"
Zen Buddhism uses a different character for zen. Unfortunately in Japanese there are a lot of words/word parts with the same way of reading but usually completely different meanings. Sometimes inflection is different, but most of the time you have to tell by context or just reading the written character.