1. Squarespace don't have the engineering to support their support team's work, therefore they need passwords. This is unlikely, but if it's the case, people should avoid Squarespace, or at least using their support system.
2. They do have the engineering support, but their support team don't know how to use it and just ask for passwords. Given how much they emphasize support, I find this unlikely, but again, if the case you should avoid using them.
3. A new starter on the support team hasn't been run through all of the details about how to use the support system yet. This strikes me as the most likely, and not grounds for avoiding Squarespace. It should be pointed out to them, privately, so that they can emphasize customer password security in their training process for support staff.
I worked at Squarespace for two years as a developer. Neither support nor engineering needs a password to diagnose and fix problems. I can't actually think of anything on the system that would require this so my first thought is that the support rep's machine or account has been compromised. Could also be an extremely poor decision on the rep's part but I never ran across another case in a couple hundred escalated issues.
I had a couple of minor styling issues once with one of Squarespace's templates. I emailed customer support and they pretty much could not have handled it any better. They were in constant contact with me and fixed both my problems (which required dev team action) quickly and totally to my satisfaction.
I was never asked for my password and I doubt very much if that's normal procedure. Without trying to sound like some shill account, I can tell you that Squarespace made a loyal customer out of me by giving me the kind of support they promise, but which I had previously assumed was sales-talk-BS.
I'm going to say it was a case of number 3 on your list.
1. Squarespace don't have the engineering to support their support team's work, therefore they need passwords. This is unlikely, but if it's the case, people should avoid Squarespace, or at least using their support system.
2. They do have the engineering support, but their support team don't know how to use it and just ask for passwords. Given how much they emphasize support, I find this unlikely, but again, if the case you should avoid using them.
3. A new starter on the support team hasn't been run through all of the details about how to use the support system yet. This strikes me as the most likely, and not grounds for avoiding Squarespace. It should be pointed out to them, privately, so that they can emphasize customer password security in their training process for support staff.