We report on the successful single crystal growth of pure EuRh${_2}$Si${_2}$ and of Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$ with $x\leq0.23$ by the flux method. Through Co substitution, EuRh$_2$Si$_2$ can be tuned from stable antiferromagnetism via a valence-transition state towards the valence-crossover regime. From magnetization measurements, we constructed a $B - T$ phase diagram for EuRh${_2}$Si${_2}$ comprising multiple magnetic phases and showing a sizable magnetic anisotropy within the basal plane of the tetragonal unit cell. This indicates a complex antiferromagnetic ground state for $x=0$. By applying positive chemical pressure through the substitution series Eu(Rh$_{1-x}$Co$_{x}$)$_2$Si$_2$, a sharp temperature-induced first-order phase transition is observed in magnetization, resistivity and heat capacity for \textcolor{blue}{0.081 $\leq$ $x$ $\leq$ 0.126}. The critical end point of this valence transition is located in the phase diagram in the vicinity of \textcolor{blue}{0.126 $<x_{\rm EDX}<$ 0.23}. At higher substitution level, the system reaches a valence-crossover regime. The obtained results are presented in a temperature-substitution phase diagram.