Negative equity capital simply means that on the Fed's balance sheet, equity (assets - liabilities) is negative. The Fed's operates on the same fundamental accounting principles as any other organization, so it has assets, liabilities, and equity ("book value"). Its total assets (such as government bonds) is actually slightly higher than its liabilities (e.g. the value of deposits by other banks), but not by much, so it's book value tends to be quite small.

The book value represents the net investment made by the owner of Federal Reserve banks.