'Does that include the right to burn one's own official US flag?'

It would seem so

The U.S. Supreme Court has made clear in several landmark rulings that speech deemed offensive to many people - even a majority - cannot be suppressed by law unless it is clearly directed to intimidate someone or incite violence, legal experts have claimed

According to Curt Anderson / Associated Press

'One applicable Supreme Court ruling came in 1989, when the justices struck down laws in Texas and 47 other states that prohibited
desecration of the U.S. flag.
That case grew out of flag-burning by otherwise nonviolent demonstrators outside the 1984 Republican National Convention in Dallas.
In a 5-4 decision, Justice William Brennan wrote for the majority: "If there is a bedrock principle underlying the First Amendment, it is that the government may not prohibit the expression of an idea simply because society finds the idea itself offensive or disagreeable." '

In another key case in 2003, the Supreme Court invalidated Virginia's law against cross-burning because it did not include a crucial component: whether the KKK intended to intimidate someone by burning the cross !