There are several related and very interesting problems and theorems:

• Schinzel's theorem - solves the problem in $\mathbb{R}^2$ using so-called Schinzel circle. 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 would be of interest (and, probably, already published somewhere)
• Kulikowski's theorem - gives explicit construction in $\mathbb{R}^3$

And similar problems related to interior points:

• Steinhaus' theorem
• Browkin's theorem

There are several related and very interesting problems and theorems:

• Schinzel's theorem - solves the problem in $\mathbb{R}^2$ using so-called Schinzel circles. 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 - gives explicit construction in $\mathbb{R}^3$ and generalizes to all higher dimensions:

W. Sierpiński, "Elementary Theory of Numbers: 2nd English Edition", 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
• Browkin's theorem