So I think I identified the issue on my site using Google's structured data validation tool. (Link to one of the impacted pages).
It appears the template I'm using (Moksha) is causing two 'product' structured data items to appear to Google for a page which should only have one, the second of which is causing this error.
<html xmlns:og="http://opengraphprotocol.org/schema/" xmlns:fb="http://www.facebook.com/2008/fbml" xmlns:product="http://ogp.me/ns/product" lang="en-US" itemscope itemtype="http://schema.org/Product">
<meta itemprop="name" content="Sensory Deprivation Tank — Indigo Aire Spa & Wellness Center"/>
<meta itemprop="url" content="https://www.indigoaire.com/spa/sensory-deprivation-tank"/>
<meta itemprop="description" content="$60 for 1 Hour Float Session $165 for three 1 Hour Float Sessions Call for package specials Choose options below to view pricing."/>
<meta itemprop="thumbnailUrl" content="http://static1.squarespace.com/static/5a1f2b5c8c56a86c49a38f90/5dbb3d8dce132b64c58d7833/5dbb42576f8b1269bc242f03/1572725664375/y.png?format=1500w"/>
The lines above are, I believe, the culprit. These <meta itemprop...> tags seem to be the second 'product' structured data item which has no 'offers, reviews, or aggregateRating' specified, and which is causing the error. I found it because one of the warnings I was getting (and still am) was about the 'product' structured data type not containing a 'thumbnailUrl' property.
I found a potential workaround, which doesn't seem to be breaking anything, but some input from someone who knows more about it than myself would be good. I haven't done any work with these structured data types before, and my practical web design knowledge is about 10 years out of date. One of the pages which Google showed this error for has validated, and a second one is pending. For some reason, the error only showed up in the search console for two of the pages, though the data validation tool showed it for all the product pages I tested.
What I did was inject these lines into the header from the products page. (Click on the products page in the 'pages' tab, hit the 'product settings' menu, and hit 'advanced', and 'page header code injection')
<div style="display:none;" itemprop="offers" itemscope itemtype="http://schema.org/AggregateOffer">
This is specifying 'offers' for the second product and thus fixing the error that Google sees on the page. As far as I can tell, these numbers aren't changing anything with the main product Google sees, and which is showing up in the search results, so they can be anything. The ones I have here just happen to be low and high price for another service offered, but since this is code injected a level up from the individual products, you can't specify the amount on an individual level anyways.
Hopefully the error I found is what is affecting everyone else, and hopefully it will help squarespace fix it so that we don't have to use such workarounds. In the meantime, someone else may find a better workaround than this, if this is indeed what was causing the problem for others. Ideally, I think, the second 'product' structured data item specified by the <meta itemprop...> tags above would be deleted completely, but I don't think that's possible to do as a user.