Industry Standard for Single-Board Computers?
Whether or not the Arduino company goes out of business isn't all that relevant. For one thing, there are now two competing companies vying for the Arduino trademark - arduino.cc and arduino.org.
The Arduino has such a ground-swell of popularity that clone boards are widely available on eBay and from other suppliers. While people are willing to buy them, someone will sell them.
The design is open-source, which makes it a safer choice than other designs who keep their design proprietary. For one thing, it guarantees that someone will make compatible boards, even if Arduino(s) fold.
The "heart" of the Arduino is (in most cases anyway) a chip from Atmel - so you really should ask if Atmel will go out of business. Probably not, but in any case, an alternative supplier might just as easily go out of business.
I wouldn't be building an "Arduino board" into a production system anyway - too much chance that wires would fall out of the prototyping sockets. However the prototyping board is good for learning about their capabilities, and testing your code.



  Do Arduinos, or any other single-board computer, come in any sort of industry-standard form factor?


I'm not aware of any industry-standard form factor. The Uno design has caused a lot of people to make "shields" that fit onto that form factor, so that is probably the closest you would get to an industry standard.