In my opinion, the maximum principle to which Bombieri, De Giorgi and Giusti refer is the same elementary maximum principle described (again without giving a precise reference or demonstration) by Miranda in [2] (§1, Theorem 1.2, pp. 667-668). And I think they all refer to the so called weak maximum principle for the area functional, as described by Giusti in [1], so in this answer I'll closely follow this reference, which also uses the notation in the OP.
The "elementary maximum principle" for area minimizers
Let $\Omega$ a bounded domain with Lipschitz continuous boundary $\partial\Omega$. Consider the following subsets of the space of Lipschitz continuous functions
$$
\newcommand{\Lip}{\operatorname{Lip}}
\begin{eqnarray}
\Lip_k(\Omega) &=&\{ f\in C^{0,1}(\Omega): |f|_\Omega\le k\} & \quad k>0 \\
\Lip_k(\Omega,\eta) &=&\{ f\in \Lip_k(\Omega): f|_{\partial\Omega}=\eta|_{\partial\Omega}\} & \quad \eta \in C^{0,1}(\Omega)\\
\end{eqnarray}
$$
and let
$$
\mathscr{A}(f,\Omega)=\int\limits_\Omega \sqrt{1+|Df|^2}\mathrm{d} x\label{4}\tag{1}
$$
be the area functional, which is strictly convex, i.e.
$$
\mathscr{A}\left(\frac{u+v}{2},\Omega\right)<\frac{1}{2}\big[\mathscr{A}(u,\Omega) + \mathscr{A}(v,\Omega)\big]\label{5}\tag{2}
$$
for each $u, v \in C^{0,1}(\Omega)$ such that $Du\neq Dv$ in $\Omega$. Then we have the following
Weak Maximum principle ([1], Lemma 12.5, p. 139). Let $f_1, f_2 \in \Lip_k(\Omega)$ respectively be a subsolution and a supersolution of the area functional minimization problem, i.e.
$$
\begin{eqnarray}
\mathscr{A}(f_1,\Omega)\le \mathscr{A}(v,\Omega)&\quad\forall v\in\Lip_k(\Omega)\text{ such that }f_1 \ge v\\
\mathscr{A}(f_2,\Omega)\le \mathscr{A}(v,\Omega)&\quad\forall v\in\Lip_k(\Omega)\text{ such that }f_2 \le v
\end{eqnarray}
$$
If $f_1\le f_2$ on $\partial\Omega$, then $f_1\le f_2$ on $\bar\Omega$ (the closure of $\Omega$).
Proof. Let's prove the result by contradiction and thus suppose that
$$
K=\{x\in\Omega:f_1(x)>f_2(x)\}\neq\emptyset.
$$
Define $\overline{f}=\max\{f_1, f_2\}$: obviously $\overline{f}\in \Lip_k(\Omega,f_2)$ and $\overline{f}\ge f_2$ therefore
$$
\mathscr{A}(f_2,\Omega)\le \mathscr{A}(\overline{f},\Omega) \iff
\mathscr{A}(f_2,K)\le \mathscr{A}(f_1,K).
$$
In the same way, defining $\underline{f}=\min\{f_1, f_2\}$, we obviously see that $\mathscr{A}(f_2,K)\ge \mathscr{A}(f_1,K)$, thus
$$
\mathscr{A}(f_2,K)= \mathscr{A}(f_1,K).
$$
Now, since $f_2=f_1$ on $\partial K$ and $f_1>f_2$ in $K$, it must be $D f_1\neq Df_2$ on a set of positive measure in $K$ therefore by \eqref{5}
$$
\mathscr{A}\left(\frac{f_1+f_2}{2},K\right)<\frac{1}{2}\big[\mathscr{A}(f_1,K) + \mathscr{A}(f_2,K)\big] = \mathscr{A}(f_2,K)
$$
but this is impossible since $f_2$ is a supersoluion in $\Lip_k(\Omega)$ and thus
$$
\mathscr{A}\left(\frac{f_1+f_2}{2},K\right) \ge \mathscr{A}(f_2,K)
$$
since $\frac{1}{2}(f_1+f_2)>f_2$. $\blacksquare$
The weak maximum principle in the paper of Bombieri, De Giorgi and Giusti
Let's consider inequalities \eqref{2} and \eqref{3} and the functional derivative of \eqref{1} on the "points" (subsolution and supersolution) $f_1$ and $f_2$:
$$
\begin{eqnarray}
\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \mathscr{A}(f_1+\varepsilon\varphi,D_1)&\le 0 & \iff \int\limits_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{1}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{1}\right|^{2}\right)^{-\frac{1}{2}} \mathrm{d}x \le 0\\
\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \mathscr{A}(f_2+\varepsilon\varphi, D_1)&\ge 0 & \iff \int\limits_{D_{1}} \sum_{i=1}^{8} \frac{\partial f_{2}}{\partial x_{i}} \frac{\partial \varphi}{\partial x_{i}}\left(1+\left|D f_{2}\right|^{2}\right)^{-\frac{1}{2}} \mathrm{d}x \ge 0
\end{eqnarray},
$$
for all $\varphi \in C^{\infty}(D_1)$ such that $\varphi\ge 0$. Inequalities \eqref{2}and \eqref{3} (whose right side is de facto the weak formulation of the minimal surface operator as shown in problem \eqref{1}) are the functional derivatives of the area functional \eqref{4} and their solutions are respectively subsolutions and supersolutions of the area minimization problem, and thus the weak maximum principle holds for them: this allows the authors deduce the double sided estimate that originated this question.
Notes
- The weak maximum principle is elementary in the sense that it does not involve any concept outside the realm of basic multivariable real analysis: the tools used for the proofs are simply order relations and the (strict) convexity of the functional \eqref{4}.
- The weak maximum principle does not requires the existence of sub/superminimizer of the area functional \eqref{4}: in practical cases, when we can effectively construct such minimizers, we can use the pinciple in order to eventually prove relevant existence and uniqueness results for Plateau's problem.
- The weak maximum principle is "weak" in the sense that the fact that it does not implies that a minimizer is the constant function if it has a minimum/maximum in the interior of $\Omega$, like the strong maximum principle for Laplace's equation does.
- I was not able to find the original reference for the weak maximum principle: according to Miranda ([2], §1, p. 668), it was used by Von Neumann to prove a maximum principle for the gradients of minimizers of the functional \eqref{4} (see the references therein for the details) and it seems that also Hilbert was aware of the result in a simpler case. It is probably due to the "fog" which seem to surround the original source that Bombieri, De Giorgi, Giusti do not gave precise references.
References
[1] Giusti, Enrico, Minimal surfaces and functions of bounded variation,
(English) Monographs in Mathematics, Vol. 80, Boston-Basel-Stuttgart: Birkhäuser, pp. XII+240, ISBN: 0-8176-3153-4,
MR0775682, Zbl 0545.49018.
[2] Miranda, Mario, "Maximum principles and minimal surfaces",
(English) Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, IV Serie, 25, No. 3-4, 667-681 (1997),
MR1655536, Zbl 1015.49028.
