The Economist has an excellent, if not terribly detailed, explanation [subscription required] of why what happened to banks will not happen to insurers.

Although some insurers do indeed have 'impaired assets' due to significant exposure to credit default swaps and other investment vehicles, unlike banks it's tough to see how there could be a 'run' on an insurance company. Here's how the Economist sees it:

"Unlike banks, which rely heavily on debt funding, insurers’ main liabilities are the claims they will pay their customers—for life firms these stretch over many years. Whereas the depositors and lenders who provide funds to banks can jump ship overnight, insurance customers find it hard and expensive to wriggle out of their contracts.

A run on an insurance company is thus hard to imagine."

Agreed.