In upcoming joint work with Jarod Alper, Jack Hall and Daniel Halpern-Leistner:

*Artin algebraization for pairs and applications to the local structure of stacks and Ferrand pushouts*

we prove more generally the existence of pushouts of affine morphisms along closed immersions in the category of (quasi-separated) algebraic stacks. This in particular implies that Milnor squares are pushouts in the category of (quasi-separated) algebraic stacks. Let me sketch how this is proved:

Let $X=\operatorname{Spec} B$ and $Y=\operatorname{Spec} A$ and similarly for the primes so we have a cartesian square:
$\require{AMScd}$
\begin{CD}
X' @>f'>> Y'\\
@V g' V V @VV g V\\
X @>>f> Y
\end{CD}
with $g$, $g'$ closed immersions.
By assumption, this is co-cartesian in the category of affine schemes.
To show that this is co-cartesian in the category of algebraic stacks, let $Z$ be an algebraic stack together with maps $u\colon X\to Z$ and $v\colon Y'\to Z$ and a $2$-isomorphism $ug'\cong vf'$. We can replace $Z$ with an open quasi-compact neighborhood of the images of $u$ and $v$ and assume that $Z$ is quasi-compact.

Let $p\colon Z_1\to Z$ be an affine smooth presentation. Consider the pull-backs along $u$, $ug'\cong vf'$ and $v$ and call these $X_1\to X$, $X'_1\to X'$ and $Y'_1\to Y'$. The easiest case is if $Z$ has affine diagonal. Then $p$ is affine and $X_1$, $X'_1$, $Y'_1$ are also affine. Then we can take the pushout of the three affine schemes resulting in $Y_1\to Y$. This gives us a map $Y_1\to Z_1\to Z$. One then observes that $Y_1\to Y$ is smooth (flatness is [Fer, Thm 2.2 (iv)] and finite presentation can be proven similarly and smoothness then follows by considering fibers). Then take $X_2=X_1\times_X X_1$ etc. We obtain two maps $Y_2\rightrightarrows Y_1\to Z_1\to Z$. Since $Y_2$ also is a pushout in the category of affine schemes (they are stable under flat base change by [Fer, Thm 2.2 (iv)]) these two maps coincide (*). By descent, we obtain a map $Y\to Z$.

(*) It remains to show that any two maps $Y\to Z$ fitting in the diagram are isomorphic up to unique 2-isomorphism. For this, one takes two maps and pull-back the diagonal of $Z$. This is then turned into an existence question. Again, if the diagonal is affine, it is immediate.

When the diagonal is not affine, then the $X_1$, $X'_1$ and $Y'_1$ above are merely algebraic spaces. One can take an étale affine presentation of $X_1$ and pull this back to $X'_1$. The crucial step is then to extend this to an étale presentation of $Y'_1$. This is where the Artin algebraization alluded to in the title comes in. It is also needed when you want to construct the pushout $Y$ of a diagram $X\leftarrow X'\rightarrow Y'$ of algebraic stacks (affine / closed immersion).

**Edit:** In [TT], the case where $\Delta_Z$ is (ind-)quasi-affine is handled. The crucial result is [TT, Thm 5.7/5.8] which in the setup above proves that $Y_1$ exists when $X_1$ is (ind-)quasi-affine. This settles the case when $Z$ is an algebraic space or a Deligne–Mumford stack with separated diagonal. The case where $f$ is finite/integral is easier and treated in [Fer] and [R, Thm. A.4]. Also see MO question Ferrand pushouts for algebraic stacks.

[Fer] Daniel Ferrand, *Conducteur, descente et pincement*, Bull. Soc. Math. France 131 (2003), no. 4, 553–585.

[R] David Rydh, *Compactification of tame Deligne–Mumford stacks*, preprint, https://people.kth.se/~dary/tamecompactification20110517.pdf

[TT] Michael Temkin and Ilya Tyomkin, *Ferrand pushouts for algebraic spaces*, Eur. J. Math. 2 (2016), no. 4, 960–983.