Could the government make knowing your DNA,and it's history mandatory? Do you care?

The US government.

Does the constitution, or any other document even come close to protecting your DNA?

The Census Bureau has the right to know how many people are in certain places, and their ethnicity. So that’s already pretty close.

For reasons I can’t really articulate, I don’t want anyone to know my DNA, or where I originated. I guess it’s a privacy issue to me.