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.
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). You can register to vote as an overseas voter here.
When registering to vote as an overseas voter, you will use the address of the last real home you had in the U.S.— where you actually resided. This is your "voting residence address", and it is this address that defines your state and jurisdiction for voting. You cannot use a P.O. Box as your last U.S. address.
For example, if you used to live in an apartment in Boston 30 years ago before moving to France to live, you would use your old apartment address in Boston for voting purposes. It doesn't matter if your parents have been receiving your mail at a different address, if you have no connection to Boston anymore, or if you have no mailing address in the US whatsoever.
Each person has one and only one residence, or domicile. When a civilian moves from one place to another, with the intent to make the new place home for a significant time, the individual's domicile changes to the new place.
For example, Jane lived and worked in Ohio, but she visited family for one month in Maine before moving overseas. She last established residence/domicile in Ohio because she was only visiting Maine and didn't make her home there. Therefore, her voting residence address as an overseas voter will be in Ohio.
Note that if you intend to live outside the United States for the long term and you aren't maintaining your domicile in your previous state of residence, under federal law you continue to maintain the right to vote for federal offices (President, Vice President, U.S. Senator, and U.S. Representative). That federal law is called The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA).
However, to be eligible to vote for state and local offices (such as Governor, State Legislator, County Clerk, Mayor, etc.), the voter must typically maintain residence/domicile in his/her last state of residence even when living overseas. For more details, you will need to check the laws of your state, which you can do so by going to the State Voting Requirements page, inputting your state, and looking under the “Eligibility Requirements” tab.
If you are serving in the uniformed services, or are the spouse or dependent of someone serving in the military, you can find information under the question "As a uniformed services member, spouse or dependent, how do I determine my legal residence address for voting purposes?" in the Address-related section of this FAQ.