Can a Unilateral Amplifier be Unstable?
It depends whether your 2-port is an arbitrary black box where you can dial in any S-parameters, or something feasible like a transistor where the values are constrained to be physical.

If the former, consider a back box with |S11|>1, and |Sxy|=0 for x,y !=1. It will be unstable for some input impedances regardless of output load, yet it's unilateral by your definition.

Computing stability circles for a port means considering the network as a whole, so for the input it involves both the S11, and the S12/21/22/load. What the ideal of unilaterality does is reduce the second term to zero. Often the |S11| is &lt;1, and then you've automatically got stability with passive inputs. But it isn't always. Look up a Pierce oscillator for instance. It's quite easy to turn an emitter follower into a Pierce with some strays.