Hilbert's Theorem states that there exists no complete analytic (class Cω) regular surface in $R^3$ of constant negative Gaussian curvature K.
For positive Gaussian curvature also when the sphere and surfaces homomorphic to it are excluded,the hyper and hypo spheres belong to the same category.They have cuspidal edges, one principal curvature goes infinite.
Why is then there no theorem about them asserted in the same way, by Hilbert or subsequent researchers?
I think, the question you meant to ask is:
"Suppose that $(S^2,g)$ is the 2-sphere equipped with a metric $g$ of constant curvature. Is it true that the image of any isometric embedding $(S^2,g)\to R^3$ is congruent to a round sphere in $R^3$?"
The answer to this is positive, proven in much greater generality by Cohn-Vossen in 1927:
Let $(S^2,g)$ be the 2-sphere equipped with a metric of curvature $\ge 0$, such that the set of points of zero curvature is nowhere dense. Then there exists at most one, up to congruence, smooth isometric embedding of $(S^2,g)$ into $R^3$.
Incidentally, a sharper version of Hilbert's theorem is due to Efimov (1963): There is no $C^2$-smooth isometric immersion of the hyperbolic plane into $R^3$. (Efimov also proved it for all complete surfaces of curvature $\le -1$.)
See also: T. K. Milnor, Efimov’s theorem about complete immersed surfaces of negative curvature, Advances in Math. 8 (1972), 474–543.
You can find much more on this topic in the book Q. Han, "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces," AMS Math. Surveys and Monographs, Vol. 130.