If $\displaystyle {X_i}$ for $\displaystyle i=1,...n$ are $\displaystyle i.i.d$, then can we show that $\displaystyle {X_i ^2}$ for $\displaystyle i=1,..n$ are also $\displaystyle i.i.d$?
yup
Whatever you had for the distribution of X, your new random variables $\displaystyle Y_i=X^2_i$
will all have that new distribution.
And also if $\displaystyle X_i$ is independent of $\displaystyle X_j$ for $\displaystyle i\ne j$
then $\displaystyle X_i^2$ is independent of $\displaystyle X_j^2$ for $\displaystyle i\ne j$