Do canonical Skolem hulls witness first order definable well-orders? Let $\mathcal L$ be a countable first order language and let $\mathcal M = (M; s^{\mathcal M} \mid s \in \mathcal L)$ be an $\mathcal L$-model.
Definition 1. $\mathcal M$ has a lightface definable well-order iff there is a $\mathcal L$-formula $\phi(x,y)$ such that
$$
\{ (x,y) \in M^2 \mid \mathcal M \models \phi(x,y) \}
$$
is a well-order of $M$.
Observation 2. Let $\mathcal L$, $\mathcal M$ be as above and let $\prec_\phi$ the lightface well-order defined by $\phi$ over $\mathcal M$. Then we can define, for any $X \subseteq M$ a canonical Skolem Hull
$$
\operatorname{Hull}^{\mathcal M}_{\prec_\phi}(X) \prec \mathcal M
$$
as follows: For each $\mathcal L$-formula $\chi(x_0, \ldots, x_n)$ and each $p_1, \ldots, p_n \in M$ let
$$
\tau^{\mathcal M}_\chi [p_1, \ldots, p_n ] =
\begin{cases}
\min_{\prec_\phi} \{ x \mid \mathcal M \models \chi[x,p_1, \ldots, p_n] \} & \text{, if this set is nonempty} \\
\text{undefined} & \text{, otherwise}
\end{cases}
$$
Then $\operatorname{Hull}^{\mathcal M}_\phi(X) := \bigcup_{n < \omega} X_0$ with $X_0 := X$ and
$$
X_{n+1} := \{ \tau^{\mathcal M}_\phi[p_1, \ldots, p_n] \mid p_1, \ldots, p_n \in X_n \}
$$
is (the universe of) the $\subseteq$-least elementary substructure of $\mathcal M$ that contains $X$ as a subset. Furthermore, if $\prec_\phi, \prec_\psi$ are distinct lightface well-orders of $\mathcal M$ then, for all $X \subseteq M$
$$
\operatorname{Hull}_{\prec_\phi}^{\mathcal M} (X) = \operatorname{Hull}_{\prec_\psi}^{\mathcal M}(X).
$$
Hence we may simply write $\operatorname{Hull}^{\mathcal M}(X)$ for this elementary substructure.
Observation 3. Let $\mathcal{M}$, $\mathcal{L}$ be as above but don't assume that $\mathcal M$ has a lightface definable well-order. Fix any well-order $\sqsubseteq$ of $M$. Then $\mathcal{M}[\sqsubseteq] = (M; \sqsubseteq,  s^{\mathcal M} \mid s \in \mathcal L )$ has $\sqsubseteq$ as its lightface $\mathcal L \cup \{ \dot{\sqsubseteq} \}$-definable well-order and it's certainly possible to have distinct well-orders $\sqsubseteq_0, \sqsubseteq_1$ of $M$ such that
$$
\operatorname{Hull}^{\mathcal M[\sqsubseteq_0]}(\emptyset)\neq \operatorname{Hull}^{\mathcal M[\sqsubseteq_1]}(\emptyset).
$$
This leads to the following
Question 4. Suppose that $\mathcal M$ is an $\mathcal L$-structure such that for all $X \subseteq M$ and for all well-orders $\sqsubseteq_0, \sqsubseteq_1$ of $M$ we have
$$
\operatorname{Hull}^{\mathcal M[\sqsubseteq_0]}(X) = \operatorname{Hull}^{\mathcal M[\sqsubseteq_1]}(X).
$$
Does it follow that $\mathcal M$ has a (lightface) definable well-order?
I suspect that the answer to this question is 'no' and that there is a quite trivial counterexample. However, since I'm apparently spoiled by the uniformity of the structures I'm typically dealing with, I seem to be unable to come up with some such example.
 A: Let $L$ be a language with constant symbols $\{c_i\mid i\in \omega\}$, and let $M$ be the structure with domain $\omega$ in which $c_i$ names $i$. Then $M$ has no definable well-order, but it also has no nontrivial substructures, so any Skolem hull of any set with respect to any well-order gives all of $M$.

Suppose $M$ is a structure with any proper elementary substructure $N$. Let $\prec_1$ be any well-order of $M$ such that $N$ is an initial segment. Then the $\prec_1$-Skolem hull of $\emptyset$ is contained in $N$. Why? For any formula $\varphi(x,\overline{a})$ with parameters $\overline{a}$ from $N$, if $M$ contains an element satisfying $\varphi$, then $N$ does too, so the least such element is in $N$. By induction, we only add elements of $N$ to the Skolem hull.
Now let $\prec_2$ be any well-order of $M$ such that the $\prec_2$-minimal element is in $M\setminus N$. Then the $\prec_2$-Skolem hull of $\emptyset$ contains this element (it's the minimal element satisfying $x=x$). In particular, the $\prec_1$-Skolem hull and $\prec_2$-Skolem hulls differ. 
Conclusion: The hypothesis of your question holds if and only if $M$ has no proper elementary substructure. And this condition has nothing at all to do with definable well-orders.
