We analyze the vacuum structure of spontaneously broken N=2 supersymmetric gauge theory with the Fayet-Iliopoulos term. Our theory is based on the gauge group SU(2) x U(1) with N-f=1,2 massless quark hypermultiplets having the same U(1) charges. In the classical potential, there are degenerate vacua even in the absence of supersymmetry. It is shown that this vacuum degeneracy is smoothed out, once quantum corrections are taken into account. In the N-f=1 case, the effective potential is found to be the so-called runaway type, and there is neither a well-defined vacuum nor a local minimum. On the other hand, in the N-f=2 case, while there is also the runaway direction in the effective potential, we find the possibility that there appears a local minimum with broken supersymmetry at the degenerate dyon point.