I’m hearing a lot of muttering about free speech rights relating to the Capitol protest/attempted coup last week. A number of people have been fired by employers after storming the Capitol, and they are complaining about their rights being violated. 

So this raises the question: Can I be fired for attending the Capitol protest?

Short answer is yes, with some limited exceptions.

But if your question is: Can I be fired for storming the Capitol?

The answer is yes, yes, a thousand times yes.

The difference is attending a protest versus engaging in clearly illegal activity. Those that stormed the Capitol are guilty of crimes ranging from vandalism to burglary to sedition to felony murder, among other things, so emloyers can definitely fire you for committing a crime. Extra bonus firing if you wore company insignia while committing the crime.

For protesters who had the sense to leave when things got crazy, there are some possible protections. 
In some states, employers’ threats to terminate employees based on politics may be illegal. Some states, like California, Colorado, New York, North Dakota and Louisiana, say it’s illegal to retaliate against an employee for their legal off-duty participation in politics or political campaigns. California, Colorado and North Dakota also protect employees from being fired for legal off-duty activities, political or not. 
Illegal activities aren’t protected at all, anywhere.

In general, remember that the First Amendment doesn’t protect you at all in a non-government workplace, and government workers have little protection in such situations.

If your protesting landed you on the do-not-fly list and your job requires travel, that’s a legitimate reason for employers to fire you even if there is some legal protection in your state otherwise.
In sum, all but a few of the protesters can be fired for attending last week’s protest, and anyone who raided the Capitol can almost certainly be fired.
Once the insurgents are convicted, most states allow employers to discriminate against those with criminal records, so that’s an extra bonus for them.