if a valuable and trusted co worker is fired on the spot, should you continue to work for that company?

I wouldn't work for a company that viewed it's employees as disposable trash.

There is every possibility that you have no idea what the reason for the firing was. And I wouldn't take the fired employee's word for whatever the reason is either. Look at your employee handbook. You'll find several things that would compel a boss to fire someone 'on the spot'. Chances are your co worker committed one or more of those infractions.

For many years I worked as an administrator of a large home care agency. I had a nurse that worked in the office who I would have easily left in charge of the whole operation should I have needed to. One day I discovered she was back dating doctors' orders. I had no choice but to stand beside her desk while she cleaned it out and escort her to her car! It would have been the same if she had stolen from a patient or falsified her time sheet.

That depends how bad you need a job, and what the co-worker got fired for. Ive had good friends who got fired, but it was for good cause. I kept working for the company

My boss was once fired for surfing porn at work.

Yeah. I didn't consider it a problem staying with that company.

It all depends on why.