Double check that you've got the power tip inserted the right way to start off, first plug the cable into the battery, then plug the battery into the jb3, and then turn on the jb3. Oh, also, before you start any of this, use the continuity setting on the multimeter to be sure that there are no shorts in the cable between power and ground (or if you don't have a continuity setting, just use the resistance setting -- if you have 0 ohm of resistance, then you have a short in the cable). Also, it is probably best on your initial testing to remove the internal batteries from the jb3.
The memorex/wallyworld li-ion batteries have protection circuits built into them. When they are short circuited, and perhaps when they are hooked up backwards to equipment, the protection circuit shuts off the output of the battery. (The protection circuit would almost certainly kick in if you had the tip inserted backwards to begin with and had batteries in the jukebox to start, since the jb3 batteries might be applying an opposite charge to the external battery.) From what I can tell from my memorex one, when this happens, the only way to get it to turn off the protection circuit and return to normal is to hook it up to the charger. The moment it is hooked up to the charger, it goes back to normal and all the charge it had before is available.
Since you were getting nothing from the battery on the voltmeter, this makes me think that this is what happened to you. Otherwise, you should have been getting some low level of voltage reading on the output. And that would explain the deal with all the lights -- it would go from essentially 0 charge (or really 0 voltage due to the protection circuit) to fully charged as soon as it was plugged into the charger.
Hopefully, this was the problem and the battery will turn out to work fine with the jb3.