I have the understanding that the outcome of measuring the same
observable twice in a row on the same state is getting the square of
If you measure the observable $O$ and get a value $o_1$, then you immediately measure $O$ again, you'll get the value $o_1$ and not $o^2_1$. This is because the first measurement leaves the system in the state $|o_1\rangle$.
Note: measurement of an observable $O$ is not equivalent to acting on the state with $O$
This is most easily seen by acting on a superposition of eigenstates of $O$
$$O(c_1|o_1\rangle + c_2|o_2\rangle) = c_1o_1|o_1\rangle + c_2o_2|o_2\rangle$$
which is not an eigenstate of $O$. But, according to the measurement postulate, immediately after measuring $O$, the state will be an eigenstate of $O$.