The first flag is for American Samoa. It is a United States territory in the South Pacific. It is not a state of the union but it is under the control of the US State Department.
The second flag is for the Vatican City. It is the smallest sovereign state in the world. It is located within Rome, Italy.
These were the two flags that I was successful to answers during an "ice breaker" last Tuesday afternoon. As a reward I was given two Milky Way bars. It was a feat as I was the only one who knew the answers to both out of a hundred people.