Yes, Vattel believed that the citizen belongs to the all-powerful State and that it is proper for the State to prevent citizens from attempting to escape to another country if the State believes that the citizen's labor is useful to the State.
And, then there's this:
§ 240. Taxes.
If the income of the public property, or of the domain, is not sufficient for the public wants, the state supplies the deficiency by taxes. These ought to be regulated in such a manner, that the citizens may pay their quota in proportion to their abilities, and the advantages they reap from society. . . ." - Vattel, The Law of Nations or the Principles of Natural Law,Book 1, Chapter 20, § 240
So, where did you think Marx got some of his ideas for communism? To Vattel, citizens were nothing more than useful tools for the State.
Americans will never adopt Vattel's crackpot ideas about citizenship or anything else. Americans believe that, by the grace of God, citizens are free!
Vattel believed in a right of expatriation. English Common Law did not permit expatriation. US law did and does.
As I mentioned, it is no wonder that the American Republic would have been based on the writings of a SWISS philosopher. Switzerland was the ONLY NATION IN THE WORLD which had overthrown a Monarchy and established a Republic.
It was the only thing remotely similar to what the American Colonies were attempting to create. I see it as natural that they would look at the organization of the Swiss government and the writings of Swiss Jurists, because no other facsimile of what they wanted to create existed elsewhere in the world, or in History.