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) and the references given therein.

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 branched 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 again a smooth Enriques surface.

Example. Let $V \subset \mathbb{P}(1^3, \; 2)$ defined by the equation $$u^2=x^4+y^4+z^4+t^4,$$ where $x,\; y, \; z,\; t$ have weight $1$ and $u$ has weight $2$, and consider the involution $\tau \colon \mathbb{P}(1^3, \; 2) \to \mathbb{P}(1^3, \;2)$ given by $$\tau(x:y:z:t:u)=(x:y:-z:-t:-u).$$ Then $V$ is naturally a double cover of $\mathbb{P}^3$ branched on a Fermat quartic surface and moreover $V$ is $\tau$-invariant. The restricted involution $\tau|_V$ fixes the eight points of $V$ given by $u=z=t=0$ and $u=x=y=0$.

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

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 quadric 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) and the references given therein.

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) and the references given therein.