The string-flip model of quark confinement is extended and applied to ${q}^{2}$q\ifmmode\bar\else\textasciimacron\fi{} $^{2}$ systems with color, spin, and flavor degrees of freedom. A realistic model for systems of two light mesons is constructed and the meson-meson interaction is investigated with the resonating-group method. Choosing the confinement strength for the confined (``hidden-color'') state is discussed in detail. We find that the interaction in the confined state often produces two-meson bound states or low-lying sharp resonances. The ${2}^{+}$ two-vector-meson systems and the K-K\ifmmode\bar\else\textasciimacron\fi{} interaction are studied in detail. The results are compared with those for other models and the possibility of observing ``hidden-color'' dominant resonances is discussed.