How to plot a surface from a system of equations?

Define an implicit region with your equations by And-combining them:

ir = ImplicitRegion[x == y*z && y^2 == x*z, {x, y, z}];

Make a 3D plot by discretizing the implicit region:

DiscretizeRegion[ir, 3*{{-1, 1}, {-1, 1}, {-1, 1}}, 
  MaxCellMeasure -> 10^-4, Boxed -> True, Axes -> True]

You can "plot" one equation as a contour and draw the mesh lines of the other equation onto it by using the option MeshFunctions.

curve = ContourPlot3D[
  y^2 == x z
  {x, -10, 10}, {y, -10, 10}, {z, -10, 10},
 MeshFunctions -> Function[{x, y, z}, x - y z],
 Mesh -> {{0}},
 ContourStyle -> None,
 BoundaryStyle -> None,
 MeshStyle -> Thick,
 PlotPoints -> 100
 ]

Here is also a compined plot that looks a bit more fancy:

surf = ContourPlot3D[{y^2 == x z, x == y z}, {x, -10, 10}, {y, -10, 10}, {z, -10, 10}, ContourStyle -> Opacity[0.4]];
Show[
  surf, 
  curve /. Line[x__] :> {Lighter@Black, Specularity[White, 30], 
     Tube[x, 0.2]}, 
  Lighting -> "Neutral"
  ]