I have two books as reference and they both say e to the power -x[n] is unstable. They take x[n] and y[n] in terms of impulse and impulse response and prove that for n=0 we have output e^-1 and for n/=0 it is 1. Then it uses the bibo stability condition for causal and stable system and prove that the system never converges and hence it is unstable.
I am not sure I understand. BIBO criterion says that is you have bounded input then your output will also be bounded, as you've seen what I wrote, from bounded input also the output of e^x(t) is bounded.
BTW in this case it doesn't matter if your system is discrete or in the continuum, either way the same BIBO condition is satisified. The difference is that the domain of the input in one is the natural numbers and on the other is real numbers.