So it does depend on what template your are switching from and which template you are switching to. Some templates are more SEO friendly then others.
As a general rule you should try and maintain everything. This includes page titles, page descriptions, URLs, image captions and image names... If you do this then you shouldn’t really see any noticeable drop.
I just recently rebuilt my website which was on version 7.0 of Squarespace and I switched to version 7.1 Squarespace. I did not see a drop in my search engine position but I maintained and preserved everything. I basically made a carbon copy of my website on version 7.1. In fact because version 7.1 of Squarespace is faster and more SEO friendly I experienced a slight increase in my position.
That is not to say that you will have the same experience. But if you plan it out correctly and make sure that everything is maintained and carried over then you should really have too much of a problem. At most you may Experian e a slight, temporary decrease in your positioning.