No, the government should not have access to your DNA, because that would be the ultimate intrusion. The government is supposed to protect the public's right to their own individual liberty. The government is not supposed to be the one that is intruding on that liberty. Taking DNA would be like living in a police state, where the government can just do everything.
Unless you're a convicted felon, you, and you alone, should have the right to your DNA. Your DNA is who you are, and no government should have the power to know everything it biologically can about you. It's a gross violation of privacy rights, and arguably, a violation of your property rights two. I do own my DNA.