You can constrain the arguments to be symbols, or you can use ReplaceAll, or us a scoping construct to "localize" the formal variables (or some combination). Not sure if this is the best way, but something to start/play with:

g[x_, y_, z_] := 
  Module[{x1, y1, z1}, Laplacian[f[x1, y1, z1], {x1, y1, z1}] /. {x1 -> x, y1 -> y, z1 -> z}]

I think you used x in place of z in your definition of g:

Laplacian[x^2 + y^2 + z^2, {x, y, x}]

Shouldn't it be:

Laplacian[x^2 + y^2 + z^2, {x, y, z}]

?

But even after that fix, I think you're still going to have a problem. When you try to evaluate g[1,1,1] it will do all of the argument replacements first, and you'll end up with Laplacian[3, {1, 1, 1}].