Quote: 1) People would call the government crazy and lose faith in them.
Doubtful, it would probably be up to a Scientific paper to publish the results, then the media would get a hold of it. Then you would spark discussion, just like it has historically sparked discussion every other time the prospect of life on other planets was brought up.
The government isn't as diabolical as everyone thinks. They really aren't that good at keeping secrets.
Quote: 2) People would go nuts, cause mayhem, and start developing crazy
theories (like the government and aliens are working together or
something absurd like that)
Those people would (Hopefully) be in the minority, and we wouldn't have to worry about them. Then again, we do have a guy who thinks the Earth is 9000 years old on our Science Committee over here. Darwin help us.