It may run for awhile but won't last.
Wait a minute, SR-71, this isn't the usual PSU question. The PSU he has is a niche product made a few years ago that allowed people to add extra power to their PCs by adding an auxiliary PSU purely to power the GPU in the form of a 5.25" bay drive. It didn't catch on, as the few products actually manufactured were not readily available, and generally build quality was a bit questionable.
As to whether the GTX 960 will run on it, that particular card consumes only 130W under full load. The PSU, assuming 70% efficiency (Just to be safe, I calculated using a lower than average efficiency.) will still output around 175W, meaning that in theory it is enough.
However, these external PSUs are getting on in years, and I'm assuming that your's is at least 7 years old. They weren't good PSUs even when they were first released, and the cards they had to power had very low TDP; The two top of the line GPUs at the time were the Nvidia GeForce 7950 GX2 and the ATI Radeon X1950 XTX, both of which only needed 450W PSUs. They weren't as sophisticated as today's cards, and thus weren't that sensitive to voltage instability or ripple, and as a result PSUs back then were much more rudimentary than now.
All of this combined means that it isn't a good idea to run a modern day enthusiast GPU off it, even though it in theory does provide enough power. It may run for now, but it very possibly will damage the GPU and stop working later on. Save up for a new, ATX PSU that will be much better at providing power and will have protection against damage to the GPU, like short circuit protection (SCP), Overpower (overload) protection (OPP), and in some cases even overtemperature protection (OTP) that your external PSU most likely doesn't have.