transfer function - Does instability make an otherwise LTI system nonlinear (or time-variant)?

I am spinning this question off from the question from johnny. Matt L. and I have had directly opposite conclusions to johnny's question.

I want to decouple the question from issues of causality and other goofy stuff.

So we have a simple first-order recursive system described with time-domain I/O equation:

$$y[n] = p \cdot y[n-1] \ + \ x[n] \quad \quad \forall n \in \mathbb{Z}$$

Of course, the Z-transform of this is

$$Y(z) = p \cdot z^{-1} Y(z) \ + \ X(z)$$

and transfer function

$$H(z) \triangleq \frac{Y(z)}{X(z)} = \frac{z}{z-p}$$

We would normally identify this as a simple and realizable LTI system with a zero at $0$ and a pole at $p$. But in the other question, there is an issue regarding linearity and time-invariance for the case when $p=-1 \ $.

For what values $p$ is this system linear? For what values $p$ is this system time-invariant?

This is, I believe, the kernel of the disagreement I have with Dr. Matt L.