Abandon Cart Email

It might be theoretically possible to design and develop your own implementation, though the cost might exceed a couple year's worth of the difference between Squarespace plans. In addition, once you get into sending emails, usually a third-party service of some kind is engaged (and therefore additional ongoing service fees), so the savings might be quite minimal. What plan are you on now? Do you use customer accounts and, if not, how would you go about getting the user's email address? How many emails would you anticipate sending per month?

Just based on my experience the feature usually pays for itself and more, then anything else you also get with the Advance Commerce plan is just a bonus. But you will need to have reasonable volumes of online sales though. By virtue of the feature you're recovering revenue that would otherwise be lost, if that revenue generates profits greater than the feature, then it's worth paying for. Plus you're potentialy adding repeat customers, so perhaps you'll also need to take into consideration the lifetime value of your customers. 

I would look at your current analytics and see how many carts are being abandoned and how many are recoverable. I would say a good rule of thumb is if at least 10% of the recoverable carts cover the cost of the Advance Commerce plan then it is worth upgrading. Abandoned cart emails are surprisingly effective so you may find that the recovery rate is greater than 10%.

Of course the effectiveness of your emails will depend on the copy of the emails themselves. But in addition to just recovering carts I've also found them invaluable for users providing feedback. Asking a question, like 'do you need any help' can help optimise your store even further to increase future conversions; like expanding which regions you deliver to or even UI improvements. 

As for third-party software, I can't think of a way that it's really possible. As @brandon says, you can't really get the users email address to send an abandoned cart email.