How do I stop a product from being published?

Stop publishing products that are in-stock
You can tell Zero Øut to not publish an in-stock product by tagging it with 'zero_out_no_show' from the product's configuration page in your Shopify admin panel.

To stop publishing all the in-stock products without having to tag them one by one with "zero_out_no_show", you can just simply hit the "Hide out-of-stock products" toggle off using the ZeroOut app. With this, all the products hidden by ZeroOut will again be visible in the store, if not configured on the Shopify admin page.