You can't hedge against the idiosyncratic risk of losing your job, but you can hedge against rising unemployment nationally. The North American Derivatives Exchange (Nadex.com, formerly Hedgestreet.com) offers options on the unemployment rate and other economic events. I did a quick search to see if any financial writer had suggested these options as part of a strategy to hedge against the risk of unemployment. So far, I haven't found any.
I'm curious also about whether the managers of staffing firms have used these options. A number of publicly-traded staffing firms had significant amounts of net cash on their balance sheets last year; did any of them use some of that net cash to hedge against an increase in the unemployment rate? If they did, they'd have even more cash now.