Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-Kodaira classification of complex surfaces, the only possible surface of Kodaira dimension $0$ or $1$ that can appear this way is an Enriques surface.
In particular, no K3 surfaces are smooth over $\mathbb Z$. Hence the K3 double cover of $E$ is not smooth. The cover is etale away from $\mathbb F_2$, hence must be singular over $\mathbb F_2$. This means $E$ most be a classical Enriques surface (a $\mu_2$-torsor rather than a $\mathbb Z/2$-torsor or $\alpha_2$-torsor).
It is possible to get some other information about $E$:
The Picard group of this surface has rank $10$. The Galois action on the Picard group must be unramified at each prime, hence trivial, so the full lattice of cycles is defined over $\mathbb Q$. Thus by the Lefschetz trace formula, $E$ has exactly $25$ $\mathbb F_2$-points.
Some other questions that might be helpful to solve this one are:
How many K3 surfaces are there with good reduction away from $2$ (and Picard rank at least $10$, and a fixed-point-free involution, etc.)?
Given an Enriques surface over $\mathbb Q_2$, what are obstructions to good reduction over $\mathbb Z_2$, other than ramification of the cohomology?
Which classical Enriques surfaces over $\mathbb F_2$ are liftable to $\mathbb Z_2$? Can something be said about the singularities and Galois representations of their $K3$ double covers?
Can we compute the discriminants of explicit families of Enriques surfaces and try to solve the Diophantine equation $\Delta=1$?
One example of an Enriques surface over $\mathbb Z[1/2]$ whose cohomology is unramified at $2$ can be constructed as the quotient of a Kummer surface with good reduction away from $2$. Let $E_1$ and $E_2$ be two elliptic curves that are either $y^2=x^3-x$ or $y^2=x^3-4x$. Let $e_1$ and $e_2$ be $2$-torsion points on $E_1$ and $E_2$ respectively. Then we can construct (Example 3.1) a fixed-point free involution on the Kummer surface of $E_1 \times E_2$, giving an Enriques surface.
Because $E_1$ and $E_2$ have good reduction away from $2$, this surface has good reduction also. $H^2$ of the Kummer surface comes from $H^2(E_1 \times E_2)$ plus the exceptional classes of the 16 blown-up $2$-torsion points. Because these points are defined over $\mathbb Q$, the cohomology classes are unramified. $H^2(E_1)$ and $H^2(E_2)$ are unramified as well, so the only ramified part of the cohomology of the Kummer surface is $H^1(E_1) \times H^1(E_2)$. Because the involution acts as reflection on $E_1$ and translation on $E_2$, it acts as $-1$ on $H^1(E_1) \times H^1(E_2)$, so that does not descend to the Enriques surface, hence its cohomology is unramified. This surface also has $\mathbb Q$-points, thus $\mathbb Q_2$ points.
Thus I cannot see any obstruction to good reduction at $2$. However, the construction certainly does not produce a smooth model of the surface over $2$.
Does this surface have good reduction at $2$?
