If it's a public company, an employee can look at the health in many of the same ways that Warren Buffet would look at it. Look at it's profit & loss statements for the last few years. If it took on debt, try to find out what the debt was used for. Look at the credit agencies' bond rating for the company. If it's not AAA, research why. Look at the company's major customers. Is it a growing marketplace?
If it's a private company, intelligence gathering is going to be harder and you often won't have good info until you actually work there. You can try to synthesize information from glassdoor, Google News (e.g. lawsuits, settlements, etc), and other sources.
>I mainly meant "startup" (i.e. not Fortune 500)
In this case, I would ask the hiring manager (often the founder) if the company is cash-flow positive. If not, ask how much "runway" is left before the company runs out of money. Some founders may push back with "I can't disclose financials, yada yada" ... maybe because of his paranoia about competitor espionage. You then have to ask yourself if you're willing to join a company with limited information. You can join a not-yet-profitable company because sometimes, it all works out. That said, the idea of concrete financial dialogue is to make the risks transparent to the employee.