Here is a partly baked idea: Hang a sheet--a light cloth--from the segment
$(0,0,0)-(1,0,0)$ along the bottom of your square,
and from the segment $(0,1,0)-(1,1,1)$ slanting above the top of your square,
and let it sag under gravity
below the $xy$-plane, pinned to these two segments.
There are very nice cloth simulation algorithms implemented,
e.g., in [Blender below][1]. It seems you could approximate the shape
of the sheet by
hanging a catenary between the points $(x,0,0)$ and $(x,1,x)$.
By increasing the lengths of the catenaries as a function of $x$,
it seems you should be able to make the surface concave throughout.
Likely the catenaries could be replaced by parabolas.
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![Cloth simulation][2]
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**Addendum**. I defer to Pietro's more precise analysis,
but just to hint toward the idea I suggested, here is a
(substantially imperfect) rendition of the sagging sheet
that (nearly) matches the boundary conditions:
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![SheetSagging][3]


  [1]: http://wiki.blender.org/index.php/Doc:2.6/Manual/Physics/Cloth
  [2]: https://i.sstatic.net/dNWHm.png
  [3]: https://i.sstatic.net/6dDGF.jpg