Edit 2: There is in fact a very simple proof that works for any locally compact group. Suppose $Z$ is an invariant point process on a locally compact group that $0<|Z|<\infty$ with positive probability. By conditioning, we can assume $0<|Z|<\infty$ almost surely. Let $X \in G$ be the random variable defined by choosing one of the points of $Z$ uniformly at random. The stationarity of $Z$ implies that the law of $X$ must be Haar measure on $G$. If $G$ is not compact, then its Haar measure is not a probability measure, and we get a contradiction.
To prove a theorem of this kind in a general setting, it is very useful to use the so-called mass-transport principle. Edit 2: Even in light of the above simple argument, I think the argument below is still instructive; the mass-transport principle has many further uses.
Edit 3: In particular, the mass-transport principle can be used to show that a point-process can't have 'too few' points: in particular, there can be no way of assigning disjoint subsets of the group to each point in such a way that some point gets assigned a set of infinite volume. I don't believe this can be done without the MTP (also, this only works with the true, unimodular, version of the MTP, it is false for nonunimodular groups).
A good reference is Probability on Trees and Networks by Lyons and Peres.
I will do things in the finitely generated case since I usually think in terms of graphs. The continuous case is quite similar, although I don't know the optimal conditions off the top of my head.
(Edit: The mass-transport principle relies on the group being unimodular, i.e., the left and right Haar measures being equal. This is automatically satisfied in the discrete case, since there both the measures are just counting measure. However, there is a biased version of the mass-transport principle for nonunimodular groups, involving the so-called modular function. This version of mass-transport is generally much less powerful than the unimodular version, but it does still suffice to prove what you want to here.)
Claim: Let $G$ be an infinite, finitely generated group, and let $Z$ be a point process on $G$ whose law is invariant under the action of $G$. Then $Z$ has either no points, or infinitely many points almost surely.
Proof: Let $F:G^2\times \{0,1\}^G\to [0,\infty]$ be a Borel function that takes as input a pair of points $x,y \in G$ and a set $\omega\subseteq G$, which represents a possible configuration of the point process. The mass-transport principle says that if $F$ is diagonally invariant, meaning that
$$F(x,y,\omega) = F(\gamma x, \gamma y, \gamma \omega)$$
for every $\gamma \in G$ and every $(x,y,\omega) \in G^2\times \{0,1\}^G$, then
$$\mathbb{E}\left[\sum_{x \in G}F(e,x,Z)\right]=\mathbb{E}\left[\sum_{x \in G}F(x,e,Z)\right],$$
where $e$ is the identity element of $G$. We think of as a rule for sending a positive amount of mass from $x$ to $y$ that depends on $\omega$, and the mass-transport principle tells us that the expected mass received by the identity is equal to the expected mass sent out by the identity.
For each $x \in G$ and $\omega \subseteq G$, let $\omega_x$ be the set of points in $\omega$ that are of minimal distance to $x$. Define $F$ by setting
$$F(x,y,\omega)=\frac{\mathbb{1}(y \in \omega_x)}{|\omega_x|},$$
letting $F(x,y,\omega)=0$ for all $y$ if $\omega = \emptyset$.
Then we have
$$\mathbb{E}\left[\sum_{x \in G}F(e,x,Z)\right] = \mathbb{P}(\omega \neq \emptyset) \leq 1.$$
Suppose for contradiction that, with positive probability, $Z$ has a finite, non-zero, number of points. On this event, there must exist a point $y \in Z$ such that $y \in \omega_x$ for infinitely many $x \in G$. By stationarity, we deduce that $e \in \omega_x$ for infinitely many $x$ with positive probability. But then we have
$$\mathbb{E}\left[\sum_{x \in G}F(x,e,Z)\right] \geq \mathbb{E}\left[\sum_{x \in G}\frac{\mathbb{1}(e \in \omega_x)}{|Z|}\right]=\infty$$
since the quantity we are taking the expectation of is infinite with positive probability. Thus, we have contradicted the mass-transport principle. $\square$
The proof of the mass-transport principle is very easy (see Lyons & Peres), but it turns out to be extremely useful. Essentially, it allows one to do 'averaging' arguments that work even in non-amenable settings where the ergodic theorem is not available. See Lyons and Peres for references to many papers in which it is used.
