How to calculate the strength of a printed object? If I'm working with standard PLA, and I want to print a box that I can stand on without any risk of it breaking, is there any good way to calculate the appropriate print settings?

I know that structural strength comes from the infill.  Knowing this, and knowing the dimensions of the box, the weight of my body, the surface area of my shoes, and the material I'm working with, is there any good way to determine the minimum infill percentage I'd want to use in order to safely bear my weight?
Strictly speaking, it is difficult to do calculations on these materials, but not impossible (I've heard about a few commercial analysis tools that do that). The FDM process (Fused Deposition Modeling) creates a product based of fused slices of material causing an anisotropic material (this means that the properties of the material are different in different dimensions). Basically, your product will be quite strong and similar in the X and Y directions, but fragile in the Z direction (layering direction). You can imagine that every layer may be a seed for cracks to grow when you're pulling at the part. 

When applying a compression load on a product like in your example, the walls need to be strong enough to hold the pressure (not all of the load as, based on the type of infill, the infill also can/should take part of the load!) and need to be of sufficiently high percentage, not only to take part of the load, but also support the walls to prevent buckling. I remember that stress calculations for buckling are difficult and require FEA (Finite Element Analysis) for more complex objects other than bars or beams. 

I think it is difficult to determine or calculate the infill percentage based on the compression load beforehand as you do not know the exact material properties and the buckling behavior. You do know that a 100% infill will give you enough strength and support, you could try to print at a lower infill, e.g. 75%, and test if that works for you.