There are several related and very interesting problems and theorems:  

* [Schinzel's theorem](http://mathworld.wolfram.com/SchinzelsTheorem.html) - solves the problem in $\mathbb{R}^2$ using so-called [Schinzel circles](http://mathworld.wolfram.com/SchinzelCircle.html). It seems intuitively clear that it generalizes to higher dimensions by slightly adjusting radius of a hypersphere so that it contains exactly the same lattice points as its section in lower dimension, but of course, a rigorous proof is needed. Indeed, there is:
* [Kulikowski's theorem](http://mathworld.wolfram.com/KulikowskisTheorem.html) - gives explicit construction in $\mathbb{R}^3$ and generalizes to all higher dimensions:

_W. Sierpiński, ["Elementary Theory of Numbers: 2nd English Edition"](http://goo.gl/kI6UZ), page 386, the last paragraph:_
> T. Kulikowski [1] has proved that for any natural number _n_ there 
exists a sphere (in the three-dimensional space), on the boundary of 
which there are precisely _n_ points whose coordinates are integers. He 
generalized this theorem for spheres in spaces of an arbitrary $\ge 3$ 
dimension.

And similar problems related to interior points:

* [Steinhaus' theorem](http://mathworld.wolfram.com/CircleLatticePoints.html)
* [Browkin's theorem](http://mathworld.wolfram.com/BrowkinsTheorem.html)