No, you cannot choose the state where you will vote. U.S. citizens living outside of the U.S. are only permitted to register and vote in the state and county where they last established residence (domicile) in the U.S. before moving outside of the country.
American citizens born overseas who have never resided in the U.S. should use the voter residence address of their U.S. parents.
If you are unsure of what address to use, you can see the question "What address should I use when registering to vote from overseas?"
To register to vote and request an absentee ballot as an overseas voter, you must submit the Voter Registration/Absentee Ballot Request form (also known as the Federal Post Card Application, or FPCA).