NEW ANSWER:

Yes there exist infinite Morse index solutions in all dimensions $N\geq 3$. For example you can take the solution in the Li Chen paper
$$
\phi(x,y) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
and trivially cross with $\mathbb{R}$ to define
$$
\phi(x,y,z) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
Clearly this continues to solve the PDE on $\mathbb{R}^3$. Moreover, by the second link (Dancer--Farina) it cannot have finite Morse index. This works for all $N\geq 3$.

SOME COMMENTS:

This example might feel cheap, but in many situations infinite Morse index comes from having a periodic or quasi periodic structure. For example, the helicoid has infinite Morse index as a critical point of the area functional. The reason for this is that the helicoid has a discrete translation symmetry.

Thus, we see that either the Helicoid is stable or it has infinite Morse index. Proof: if the Helicoid is unstable, this means there is a compact piece that's unstable (stability is always considering compactly supported variations). Now, we can take this unstable piece and translate it sufficiently many times upwards to be disjoint from itself. This gives two $L^2$-orthogonal destabilizations. We can repeat this to get arbitrarily many. Thus the Morse index must be infinity.

Of course it *could* a priori happen that a solution with symmetries is *stable* (the helicoid is not, but this is just due to the specifics of the problem). For example, the flat plane *is* a stable minimal surface and it of course has lots of symmetries.

This is not to say that all infinite Morse index solutions have a discrete symmetry. In fact, one would expect that in general, infinite Morse index solutions can be very disordered. However, a lot of examples that we can cook up do have some symmetries (maybe this is more due to our limited knowledge than any deep truth). For an example of infinite index with only quasi-symmetries see e.g. the genus 1 helicoid.

EDIT: As pointed out by Willie Wong, the original answer considers the wrong question.

There are finite Morse index solutions for $N\geq 10$. See remark 1(i) in the second paper you reference.

[...] for every N ≥ 10 the equation (1.1) possesses a radial stable solution. The
existence of such a solution is a consequence of the analysis performed in [12], as
was remarked in [6].