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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.
As alluded to in 0scar's answer, it is important to stick to multiples of your nozzle width for thin shells. If you try to print a 1mm wall with a 0.4mm nozzle, you give the slicer a bit of a challenge in filling the joint between the two outer faces. Maybe it can print a half-width line, by under extruding, maybe not - but a 0.8mm wall is likely to be stronger.
So long as the slicer has more than 3 nozzle widths to play with, it is able to add diagonal fill to build up a solid form (as hinted in this comment on github for cura)
The answer to the question depends completely on the rigidity required. A single (vase) wall can work, particularly for a curved object - but you might find that a stronger lip is required. Although 2 or 3 perimeters will flex, they may be strong enough for your application.
Above this, you may be better off with a 2-sided wall. 2 perimeters each side with a 1mm gap (and a bit of infill) is comparable to a 6 perimeter solid wall due to the increased overall thickness and the compression/extension multiplication. I rarely go above 3 perimeters - if I need extra strength, I resort to punching holes like the slots in this part.
I make small objects (25mm^2) with 1 to 1.5 mm walls and larger objects (think coffee cups) with about 2.5 - 3 mm walls. I set the line width and number of perimeters to completely fill the thickness. I use this for ABS and PLA. The PLA objects have been electronics enclosures, with internal structures to support the parts. They aren't subjected to strong forces. I've made small containers for stain gauges with the amplifiers, and larger ones for greater sequences and Arduino boxes. I've used ABS also for coffee mugs and beverage glasses, to withstand the mass and temperature of drinks. It can take some fussing to worry the Slic3r slicing parameters into place. Especially with mugs the design is for function and appearance, not for effect.
