Also relevant should be the question how to act in different phases. An unhealthy company is not necessarily dying. And even a dying company is not necessarily bad for you. It's like with real people. When someone dies some others start to check out the valuables to get the best for themselves. If you are working in a brilliant team inside a dying company, you may all get picked up, get a raise, and be welcomed into new arms. That's one way to get into Google for instance.
For figuring out the current health status, I'd check:
the product line - is it understandable? is it modern? is it efficient?
the customer base - do they have customers that wouldn't easily change to alternative options?
the management team - do they have visions? are they cooperating? are they lying psychopaths, ambitious inventors, calm survivors (thinking Merkel here), idiotic burocrats?
HR - HR is managements comm channel to the employees. Does the promo material look good? How close is the promo material to the actual day-to-day work?
People - are there smart people you like to work with? How many of them are currently joining? How many of them are currently leaving?
Hiring - you are either new and just got hired or there for a long time and probably at least hear things about the hiring process at the water cooler. how reasonable does it sound? does it filter out idiots? does it assess quality attributes like culture? Does the feedback from the interviewers have influence on the hiring decision (more often than you think they actually just hire anybody, if they are hiring at all).