Move y-axes label beneath below y-axes?

Plot[-x, {x , 0, 10}, 
 Epilog -> (Style[Text[##], 16, "Panel"] & @@@ {{"x", {10.75, 0}}, {"y", {0, -10.75}}}),  
 PlotRangeClipping -> False, ImagePadding -> 20]

Try this:

Show[{
  Plot[-x, {x, 0, 10}, PlotRange -> {{-0.1, 11}, {-11, 0.3}}],
  Graphics[{Text[Style["y", 16, Italic], Scaled[{0.06, 0.044}]], 
    Text[Style["x", 16, Italic], Scaled[{0.99, 0.9}]]}]

  }]

Have fun!