How to determine shell thickness for parts? Let's say you're printing a container or some other object with no infill. What's a good rule of thumb for how thick to make the outside? I'm looking for something along the lines of millimeters thick per square inch of area.

I'm thinking about PLA right now, but answers for ABS and other materials are welcome, too.
This totally depends on what you think is acceptable for your print, and what the usage of the print is: is it a structural/functional part or just for aesthetic purposes. There is no general rule of thumb for that. Basically this is a design issue.

If it has to be stiff and may not flex much you would require to use more walls to give it more rigidity. Less if it is not that important.

When using infill and depending of the part, 0.8 to 1.2 mm walls are frequently used for nozzle sizes of 0.4 mm, without infill you could use the same values, but need to take less rigidity for granted. Add more walls when you need it to be more rigid. Note that the larger the wall surface area becomes (height over width), the easier a wall can buckle under load; increase the wall count to counteract.

In your case (design of a container) I would consider what the purpose of the container is; e.g. how big is it, how much weight is it going to contain (e.g. heavy nuts and bolts, or left over filament pieces), is it stackable, etc. etc. etc.. As a final thought, note that you can play with the design as well, a straight box/container is less stiff than a box with some curved edges or stiffeners. An example of a stackable container is e.g. this:



The indents on the walls of this container give it more rigidity, these walls are in the range of 2 to 3 perimeters.