According to limited but hands-on research, MFBOXCO finds that the humble Meniscus is by far the most common lens in the box cameras featured here.
Meniscus lenses are extremely simple – just a piece of optical glass or plastic with curved outer surfaces (fig. 1). The shape of the two surfaces – concave or convex – determines the focal point distances.
For example, a magnifying glass has two convex surfaces. The reason you move the glass back and forth in front of your eye is to find those focal points - the distance from glass to subject and the distance from your eye to the glass.
In the case of box cameras, the distance from lens to film is known. The focal point in front of the lens is also known - hence the 6ft to 8ft minimum focal distances of most box cameras. With these two distances known and some old-fashioned math, one can figure out what shape the curves need to take.
In order to cast the biggest image circle with the smallest and often, cheapest, lens, many manufacturers used the Reverse Meniscus design (fig. 2). This type of lens is concave on the front and convex on the back and a large image area relative to the size of the lens.
A typical box camera lens is also very often located on the INSIDE of the camera, behind the shutter (with some exceptions, of course). They are either attached to the same internal divider as the shutter base plate, or attached to the film holder itself. This allows for very simple and rugged camera designs that offered little chance for the lens to become damaged, and which didn't need a cap since the closed shutter covers the lens at all times.
Since box cameras were for the masses, it seems very unlikely that optical accuracy and consistency were all that important in the manufacture of the lenses. In today's terms, it's hard to compare, but their closest kin would be found in Holga, Diana and other "toy" cameras.
This of course means that the user may see vignetting, distortion and other cool effects (uh, we mean "optical aberrations") in their images. Many box camera lenses exhibit a general softness except in a very specific "sweet spot" - a great feature.
MFBOXCO does it's level best to discover the nature of the lens or lenses on each camera we profile. Information is, however, often limited - so please feel free to contribute your personal knowledge in the comments on each camera page.