They should, if the company folds!
They should be fired by natural causes,
Like the company breaking up under their feet and the shareholders and the rest of the board crying for their heads and getting it too!
But that can be left to adequate corporate governance and it's right because it's generally NOT YOUR MONEY. Unless you hold shares of that company. That means private. Private profits and private risks and private losses and private responsibility.
That happens all the time to directives of companies that fold and cannot afford to have lobbies in Washington to make the government willing to shower money on them whenever the going gets tough.
The question of why these people should be fired (or not), and the general public giving an opinion on this question is only accessory to the question of the car companies getting huge bailouts in federal money AT TAXPAYER EXPENSE. Like it was when the investment banks got huge bailouts at TAXPAYER EXPENSE.
So now, we accept the purest of thieveries, in the name of "Saving the Economy"...
It would not even arise in a free market because in a free market you would be either public or private and the difference well-defined. Private, you take responsibility for your financial solvency. Maybe in a freer economy, there would be more than three automakers, and the pain of letting one go would not be so great.
You know, the executives of folding auto companies would be fired or would have a huge smudge in their record, and that would be it for them. No huge bonuses, and maybe even some suits.
If the automaker is bought out by a foreign firm those who took good decisions would generally be recognized as competent and kept.