A plan with Code Injection. On Squarespace's current plans that's Core, Plus or Advanced (Business or Commerce on older plans). The free trial includes it, so you can set it up before you choose.
Use it exactly as it is. Don't change the address or download the file: the chat updates itself from there.
You only paste it once. Colours, messages and everything else change from the dashboard and reach your site within about a minute.
Add the code with Code Injection
In your Squarespace site, open Settings › Advanced › Code Injection. (Menus move from time to time: if you can't see it, search the Squarespace help center for "Code Injection".)
Paste your install code into the Footer box. Code here goes just before </body> on every page.
Choose Save.
Use the Footer box, not the Header one: your page shows first, and the chat button joins it a moment later.
The chat on some pages only
Keep the code in the site-wide Footer box, then list the pages that shouldn't show the chat under Chat Widget › Behavior › Pages without the chat, for example /members/*.
If the chat doesn't show on Squarespace
You're logged in and looking at the editor. Open your site in a private window, the way visitors see it.
Your plan doesn't include Code Injection. The box is locked or the code doesn't run. Check your plan under Billing.
The site is still private or behind a password. The chat shows once you're in, but our installation check can't get past the password.
Checkout pages. Squarespace doesn't run custom code there, so the chat doesn't show on them.
Check that it works
Open your site in a private (incognito) window, so you see it the way a visitor does.
Look for the chat button in the bottom corner of the page, then send yourself a test message.
In the dashboard, the message appears in your Inbox as it arrives. Reply, and watch the answer reach the website.
You can also choose check the installation on the Install tab of Administration › Chat Widget: we load your home page and look for your code. No button? Go through when the chat doesn't show.
Set up the rest
The chat is on your Squarespace site. A few minutes in the dashboard makes it yours: