I am given $X_t^2$ to be checked, where $X_t$ is Poisson process. How do I check? What are the properties that need to be checked?
Let $Y_t$ be a Poisson process. For what values of $k$ is $P(Y_t=k)\ne 0$?
For what values of $k$ is $P(X_t^2=k)\ne 0$?
Added: The above hint leads to probably the simplest way to see that $X_t^2$ is not a Poisson process. However, almost all of the characteristic features of the Poisson process fail for $X_t^2$.