It's not so much that you cannot renounce your citizenship, but it is severely discouraged. After some high-profile cases of wealthy individual renouncing citizenship, congress passed laws to make sure that if you want to renounce your citizen ship you need to have paid all of your taxes. Additionally, you may need to pay an additional "exit tax" if you have had significant income or have significant wealth.

As far as becoming another country's citizen, that gets to be fairly complex. Some nations recognize dual citizens, others do not. How that exactly works out depends on the particular circumstances. In any case, the U.S. generally likes to keep its citizens within its grasp.