Square does not integrate with Shopify
Square says so itself. Which half of the question you meant decides whether that matters.
Search this and the first thing you meet is Square’s own comparison page stating there is not a Square Shopify integration, with a developer forum thread saying the same underneath it. Both are accurate and both are answering the payments question: you cannot drop Square in as the card processor at a Shopify checkout without building it yourself, and that has not changed.
The stock question is a different question with a different answer. Square’s catalogue, its counts and its orders are all available to an app you authorise, which is what makes one shared stock count across the counter and the website an ordinary thing to set up rather than a workaround. That half is what this page is about.
So if you came here to run Shopify payments through Square, nothing here helps, and it is better to read that now than three apps later. If you came because the shop floor and the website disagree about how many are left, that is the problem this solves.