No.  A counter example: Let  {Xt: t ∈ Z}  be iid with Student-t(3) distribution. Then  {Xt: t ∈ Z} is weakly stationary. Howerver, Var(X²t) does not exist. Then  {X²t: t ∈ Z}  is not weakly stationary.

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A process is stationary if it satisfies the following three properties:

1. EXt exists and does not depend on t.  In our example, EXt = 0.
2. Var(Xt) exists and does not depend on t. In our example, Var(Xt)=3.
3. Cov(Xt, X_(t+h)) does not depdend on t. In our example Cov(Xt, X_{t+h}) =0 if h is not equal to 0.

So, we can conclude that  {Xt: t ∈ Z} is weakly stationary. Howerver, Var(X²t)  =EX^4t -(E(X²t))^2.  For, t(3) distribution, it does not exits fourth order moments.