Questions about analogy between Spec Z and 3-manifolds
The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields" (arXiv:0107210): the author states the Poincare conjecture as "S3 is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.
The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.
Edit: An article by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."
I think it's important to keep track of the fact that the analogy isn't between individual number fields and individual 3-manifolds; it's between the collection of all number fields and the collection of all 3-manifolds. So in my opinion it's slightly awry to ask for an "arithmetic Poincare conjecture" about Spec Z; I don't think Spec Z should be thought of as analogous to S^3 in any meaningful sense.
As always, John Baez has useful things to say.
I saw Deninger give a beautiful talk about his point of view on this, some of which is recorded in this paper. Part of the idea, somewhat vaguely, is that you should think of a number field not as an unadorned 3-manifold but as a 3-manifold with a flow on it. And then the finite primes are not just knots, but closed orbits of that flow! That gives a more satisfying answer to "why should a 3-manifold have a distinguished countably infinite family of knots on it," makes the connection with dynamical zeta functions, etc.
This has been well-addressed by the answerers before me, but just to chime in -- there are a variety of analogs one could make for the Poincare conjecture for number fields. For one, there are several equivalent statements about the Poincare conjecture for 3-manifolds which are not equivalent when transferred over by analogy to the number field case. As a first easy example, while 3-manfiolds enjoy a clean Poincare duality, number fields have extra 2-torsion. In particular, one frequently has $H^1(\mathcal{O}_K,\mathbf{G}_m)$ trivial with $H^2(\mathcal{O}_K,\mathbf{G}_m)$ non-trivial (example: any real quadratic number field with trivial class group). The equivalences (or lack thereof) between being an integral homology 3-sphere, a rational homology 3-sphere, and a homotopy 3-sphere are not the same in the two "categories." So depending on how you phrase your analogous Poincare conjecture, you may get different answers. The cleanest form (found in Niranjan Ramachandran's "A Note on Arithmetic Topology", which deals exclusively with this question) is that there are exactly ten rational homology 3-spheres which are homotopy 3-spheres, namely the 9 quadratic imaginary number fields of class number one and $\mathbb{Q}$ itself. (Or really, $\mathbb{Z}$ itself), and even more homotopy 3-spheres.
A second frequently under-emphasized point to make is that no one really knows what the right category for this analogy is on the number theory side. As mentioned above, if you take your category to be Specs of rings of integers in a number field, you don't get the Poincare conjecture. On the other, if you take the point of view of Artin-Verdier theory (or alternatively, Arakelov theory), where you include in your spaces some information about the behavior of the infinite primes (from the point of view of number theory, defining Spec(Z) as the set of prime ideals ignores the obviously important primes at infinity), then you get a different cohomology theory. With these new cohomology groups in place, some things look a little bit cleaner. Again, see Ramachandran.