I ran out of brake housing... OK to use dérailleur cable and housing for my rear brake until I can get some?

Thanks...

No

I've tried it. The cable to too thin to be secure in the clamp on the caliper.

^^^ That isn't the real issue. The housing he wants to use will literally explode if he had to brake hard at that wheel.

I don't think I would do it. I don't know about 'exploding' but I don't think it will last long.

standard brake housing is a coil, Index gear housing is a linear bundle , 3rd niche ..
there is a kevlar braided oversleeve on housing for some of the cable-disc installations that combined the not compressible feature of the SIS housing, but resist expanding like barrel staves when squeezed between brake lever and caliper ..

If you really must just to get through the next couple of days and this is your only transportation then sure, it'll work. This is the rear brake, after all, so it's not like it's your main brake. The limitations that the others have warned about are very real. Just don't be pulling on that lever like you're trying to stop the world in its orbit and you'll be OK. So no panic stops.

In other words if you ride in nasty traffic that can be trusted to do the unexpected in its attempts to prune you from this existence then I think I'd let the bike sit for another day. But if you commute on pathways and have more control over your reactions then it would work just fine.

I would do it. From Sheldon Brown:

"Warning: Since compressionless housing relies on plastic to hold it together, it is not as strong as conventional spiral housing, and should never be used for brakes! The loads applied to brake cables can easily cause compressionless housing to rupture and burst, causing a complete and sudden loss of brake function."

I'm not sure if you want to use just the housing (not a good idea as explained above) or both the housing and inner wire. A derailleur inner wire won't work either as the end that anchors in the lever is too small to work with brake levers.

Short term, in an emergency, a brake with the wrong cable and housing beats no brake at all, so if you were on the road and needed an emergency fix until you got to a bike shop it beats nothing, but that's about the only reason to do so.

brake inner wires are more than twice as strong as gear inner wires, and the coil-on-coil of spring wound is designed for the high compression loads of braking, and works fine even without the plastic cover. Index housings long spiral depends on the strength of the outer cover to support it under compression, and will buckle if the load is great as in a hard braking.

BTW- given that index housing is more than twice as much as brake housing, why would you even consider it?

I don't think "explode" is too strong a word.....

And the cost of the cable that would not be useable afterwards as well. All in all I agree that while it would be better than NO brake I'd be inclined to just take a bus for one day and buy the stuff you need to do the job right.

Or the obvious BF answer is that you need to not only stock up on such consumables as housing and cables but to also shop for a second bike as a "spare" for just such situations where you find yourself fresh out of cables and housings at midnight.....