Not really. Taxes belong to the country.
We all earn our money by virtue of the services and environment that the country provides. The electric grid, the interstate system, dams for electricity, street lights, clean air and water, government agencies that handle car registrations and deeds and records, etc.
Every citizen contributes to the country proportionate to his or her income. We pool that money for us to use to support the continuance of our country and the government, keep the environment clean and protected for future generations and enable us to make a living and pursue a free, happy, healthy life.
True to the nature of human beings, some are selfish and don't want to acknowledge the community that contributed to their success or their duty to contribute to the community, like the rest of us. Ironically, it is often those with the most who resent paying their fair share.
Our country doesn't ask much of the wealthy. Most pay a lower percentage of their income to the country than the middle class. And some very wealthy people and corporations pay nothing at all, legally.
So, no...it's not "their" money in the sense that they are not entitled to dictate how the country sees fit to use it in the best interest of the country. I may disagree with the use of my taxes, such as for a war I don't support. But we don't get to dictate specifically how each dollar of our taxes can be used. Which makes sense.
All civilized societies provide a safety net for those at the bottom, for our most vulnerable. All civilized societies provide health care to all of its citizens. That's the least of which a country should do for its citizens. If someone disagrees, maybe they should move to another country and see if their success can continue over there. Of course, if they move to another country, they'll have to look for one that doesn't provide the basics to its citizens. There aren't many successful, civilized countries that don't do that. Except here in the U.S., where people die for lack of health care.