Let me show that the answer to your last question is **yes**, by explaining how to construct an Enriques surface inside a simply connected threefold. 

Let $\psi \colon V \to \mathbb{P}^3$ be a double covering ramified on a smooth quartic surface and let $\tau \in \textrm{Aut}(V)$ be an involution fixing eight points. Set $X=V/ \tau$. Then one proves that $X$ is a *rational* threefold with eight singular points of type $\frac{1}{2}(1,1,1)$. The anti-canonical divisor $-K_X$ is not Cartier but we have 
$-K_X \equiv_{Q} H$, where $H$ is an ample divisor on $X$ with $H^3=8$ such that the general element of $|H|$ is a smooth Enriques surface.

Now consider any desingularization $\pi \colon Y \to X$. Since $X$ is rational, $Y$ is a smooth rational variety, hence $\pi_1(Y)=0$. Moreover $X$ has only isolated singularities, so the general element of $|\pi^*H|$ is a smooth Enriques surface.

For the details, see [I. Cheltsov, Rationality of Enriques-Fano threefold of genus five, Isvestiya Math. **68** (2004)][1] and the references given therein.   
 


  [1]: http://www.maths.ed.ac.uk/cheltsov/pdf/izv04b.pdf