The combination of non-minimal couplings to gravity with the post-inflationary kinetic-dominated era typically appearing in quintessential inflation scenarios may lead to the spontaneous symmetry breaking of internal symmetries and its eventual restoration at the onset of radiation domination. On general grounds, the breaking of these symmetries leads to the generation of short-lived topological defects that tend to produce gravitational waves until the symmetry is restored. We study here the background of gravitational waves generated by a global cosmic string network following the dynamical symmetry breaking and restoration of a U(1) symmetry. The resulting power spectrum depends on the duration of the heating process and it is potentially detectable, providing a test on the existence of non-minimal couplings to gravity and the characteristic energy scale of post-inflationary physics.