Did your cousin post it on their own account, or actually on your account?
This matters a lot. If it's just them tagging you in a post on *their* account, you're basically in the clear legally. Being tagged doesn't make you responsible for what someone else wrote. The business owner would go after your cousin, not you. Tagging is just a notification thing - it doesn't mean you endorsed or participated in creating the content.
Now, if they somehow posted it *on your actual account* (like they got your password or you let them use your phone), that's messier. You could potentially face liability because it's on your profile, even though you didn't write it. The business might argue you're responsible for what appears on your account. That said, most defamation cases need to prove actual damages and that you knew or should've known the statement was false. Just having false info on your wall doesn't automatically mean you're liable, especially if you can show you didn't post it and removed it quickly once you found out.
The threatening to sue everyone tagged thing is mostly bluffing on their part. Lawyers generally won't waste resources suing someone just for being tagged. But if you're actually worried, don't panic - don't delete anything or try to cover it up, just document that you didn't post it. If they actually follow through with legal action (which is unlikely), that's when you'd want to talk to a lawyer, not a Q&A site.