I'll try to answer your question "how does one determine if it is an eigenfunction of a stable, discrete-time LTI system?" for the type of sequences specified in your question. Let's consider the signal $x[n]=5^nu[n]$ and let $h[n]$ be the system's impulse response. If $x[n]$ is an eigenfunction of the system, then the output signal must be a scaled version of the input signal: $y[n]=c\cdot x[n]$
The convolution sum is
We can make the sum in $(1)$ zero for $n<0$ by requiring $h[n]$ to be causal, i.e., $h[n]=0$, $n<0$. With this requirement we get
This almost looks like the form we're looking for. However, note that the multiplicative term in $(2)$ is not constant but it depends on $n$. So $(2)$ shows that the given $x[n]$ is generally not an eigenfunction of an LTI system. Of course it is an eigenfunction of the trivial system with impulse response $c\cdot \delta[n]$, but any other sequence is so too.
You can use a similar argument to show that also the second signal in your question is not an eigenfunction of any LTI system other than the trivial system mentioned above.
In sum, if you're looking for functions/sequences that are eigenfunctions of all LTI systems, then you'll end up with complex exponentials, as pointed out in RBJ's answer. For special systems there are also other eigenfunctions, such as the ones pointed out by Jazzmaniac in the comments. E.g., any band-limited function is an eigenfunction of an ideally frequency-selective filter with a pass band extending over the frequency range of the input signal. But such ideally frequency-selective systems are not stable, so they're not in the category of systems you are looking for. Note that there are other stable LTI systems with specific eigenfunctions different from complex exponentials. But in general it is not possible to construct an LTI system for a given function such that this function becomes an eigenfunction of the system (if we exclude the trivial system $h[n]=c\cdot \delta[n]$ mentioned above). And this last sentence is also true for the two signals given in your question.