Can a causal system generate a non-causal signal or vice versa?
Can a causal system generate a non-causal signal[?]


If a non-causal signal is input to a causal system, then the output can be a non-causal signal. However if the input is zero from \$t=-\infty\$ up to some time \$t_s\$ (\$t_s\$ being less than zero to make the input signal non-causal), then (assuming we're talking about systems that are also linear) the output of the system will also be zero up to time \$t_s\$.

In practice, the concept of a non-causal signal is not particularly useful and if I've ever heard of it before, I've had no need to remember it. 


  or vice versa?


As far as I know, a signal cannot generate a system, so the answer to this is no.