Working with Squarespace Customer Support, we found the answer. The Mailing List search tool does not work with uppercase letters. Searching for "John" will yield no results, while "john" will—even if the subscriber record has it as "John." It's weird, but workable—as long as you remember this quirk every time you search your mailing lists.
Well, I found my own answer. But it's not a very satisfactory one!
Squarespace on iOS arranges products in alphabetical order. It's impractical, but you can rename your products alphabetically to force the order you want. However, if you use numbers, it will sort them as Product 1, Product 10, Product 11… instead of Product 1, Product 2, Product 3…
It would be nice if you could drag your most important product to the first slot, and arrange them the same way you can in your online store! Better yet, it would be intuitive if your product gallery in the Squarespace iOS POS tool would just duplicate exactly how you have it in your online store [sigh]!
Squarespace Help tells me they're sending my request to the development team. So...it should be a matter of days if not hours before they implement this fix. 😉