Will any type of solder bond to any non-conductive material?
"bond" is a strong word (pun intended).

So, the excellent bonding properties of let's say solder-copper are basically because solder partially migrates ("solves") into the structure of the metal (and the other way around), so that you basically get interwoven/interlocking crystals.

That will only work with metals. Metals conduct.

So, the next best thing is gluing behaviour. And that can already be pretty strong between any heat-resistant, smooth surface and solder. You'd need very clean surfaces, and something to make the surface creep work for you. It seems (thanks, Jonathan!) that ultrasound can help with that :)

Another option is actually going the more destructive road: If trying to solder something onto a thermoplastic plastic part, then that part will simply melt around the piece, and that can be an excellent plastic weld. Superior quality welds between metal and plastic is, however, usually achieved through ultrasonic punctual heating (see: joghurt lid made from aluminium). 

Then, it really depends on what you need to achieve, but: 

You could also use one of the chemical deposition methods used to coat non-conductors (plastic, ceramics) in copper (and often, after that, ultimately in chrome) and plainly solder to that.

Maybe, also a polished surface of a plastic with copper pieces embedded would have enough copper surface for mechanical bonding, but due to a lack of touching copper, no conductivity.