Do citizens have a duty to speak out or shut up?

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Do free citizens in a democracy

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Contradictory. Democracy is inherently freedom-limiting.

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have a right to object to things their government does that the believe are offensive, immoral or illegal?

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This may or may not be a right that the democracy does not violate. The people have this right whether or not the democracy (or any other form of government) recognizes it.

The more interesting question is "do these people, who support and participate in the democracy, have a *legitimate* reason to complain when the system produces outcomes they don't agree with?"

We don't tolerate those who whine about the Red Sox beating the Yankess for more than, what, a week? But those who whine for years about politician XYZ beating politician ABC are encouraged. Hey, you voted, your team lost. That's how it goes, so shut up. You *more* than tacitly consented. Save your bluster for the next election.

How hypocritical do you have to be to say "I support democracy, but only if I get the results I want"? Or "I support democracy, but I want to arbitrarily definie who gets to vote with a vague "civics" tests"? Or "I support democracy, but I want to impeach this guy even though I can't figure out what he's done that's impeachable under the rules that I ostensibly agree with"? Interestingly, people who advocate the last position would likely fail the arbitrary civics tests proposed by those that advocate the second position (and sometimes they are the same person!).

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Do they have a duty to do so? Or do they have a duty to keep quiet? Or is the right qualified by wartime?

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If they have a duty, then they are not "free" citizens as you specified. But we already knew that was a misnomer, since you then specified that they were living under a democracy.