Counter-example to show that this system is non-linear?
OK. Let's try homogeneity again:
$$
y[n] = x[n]\cdot x[n-1]
$$
for input $x[n]$.

For input $a\cdot x[n]$ we get
$$
y'[n] = a\cdot x[n] \cdot a x[n-1] = a^2 \cdot x[n]\cdot x[n-1] \not = ay[n]
$$
so homogeneity doesn't apply.

Let's try that with an example, as per MBaz's suggestion.

Let's try:
$$
x[n] = 2
$$
so that
$$
y[n] = 4
$$
Then let's try twice that:
$$
x'[n] = 2\cdot x[n] = 4
$$
So that
$$
y'[n] = 16 \not= 2\cdot y[n]
$$