If the government really wants your DNA it isn't hard to obtain legally. They'll just follow you around everywhere until you leave behind your DNA and then grab it. They'll wait until you drop your coffee cup in a public trash or spit on the ground or leave behind a cigarette butt. Many crimes have been solved this way.
For example see COMMONWEALTH v. Jeffrey BLY http://caselaw.findlaw.com/ma-supreme-judicial-court/1186295...
>Suppression issues. (a) Physical evidence. Bly argues that the method used by the Commonwealth in obtaining his known DNA sample constituted a nonconsensual seizure and thus violated his rights under the Fourth and Fourteenth Amendments to the United States Constitution, under art. 14 of the Massachusetts Declaration of Rights and under the search warrant requirements of G.L. c. 276, § 1. The judge denied Bly's motion to suppress on this issue, finding that the cigarette butts and water bottle seized by the police constituted trash that was abandoned by Bly.
Or STATE OF WASHINGTON v. JOHN NICHOLAS ATHAN http://www.denverda.org/DNA_Documents/Athan.pdf
>We find there is no inherent privacy interest in
saliva. Certainly the nonconsensual collection of blood or
urine samples in some circumstances, such as under the
facts of Robinson, invokes privacy concerns; however,
obtaining the saliva sample in this case did not involve an
invasive or involuntary procedure. The relevant question
in this case is whether, when a person licks an envelope
and places it in the mail, that person retains any privacy
interest in his saliva at all. Unlike a nonconsensual
sampling situation, there was no force involved in
obtaining Athan's saliva sample here. The facts of this
situation are analogous to a person spitting on the
sidewalk or leaving a cigarette butt in an ashtray. We
hold under these circumstances, any privacy interest is
lost. The envelope, and any saliva contained on it,
becomes the property of the recipient.