rovnice[y0_]:={y''[x]+(y'[x])^2+y[x]==0,y[0]==y0,y'[0]==0};
reseni[y0_]:=NDSolveValue[rovnice[y0],y,{x,0,20}];

reseni[0.4]

reseni[0.4][2.4]

Timing[Plot[reseni[0.4][q],{q,0,20}]]

Timing[Plot[Evaluate[reseni[0.4][q]],{q,0,20}]]