Can a pogo pin spring cause an inductive spike?
It's hard to say without a bit more info on the layout, currents, operation etc. However, the best way to figure this out would be to see if you can generate a spike and capture it on a scope in single shot mode (shouldn't be too difficult with a DSO)  

Most modern uCs have input protection, so it would have to quite a hefty spike to do damage like you mention, but it can certainly happen to just one pin and not any others.  

I would try the above (even test just with a pogo pin on it's own and switch from 3.3V to 0V and monitor.  

Also, is there any extra capacitance on the pin that could resonate with the inductance? What does the pin do normally - does it have standard input protection if it's an external I/O?
Posting the relevant part of the schematic would be useful (just enough to figure out any possible problems whist keeping the design private)