Squaring the norm is basically squaring (hence increasing) the condition number, which essentially takes away digits of accuracy in the solution process, hence making it less numerically stable.
If the norm (or its square) is the only term in the objective function, then the argmin (if exactly computed) will be the same for the squared or not versions. That is true whether or not there are any constraints. However, if there is an additional term in the objective function, then squaring or not the norm term can change the argmin.
Please read http://cvxr.com/cvx/doc/advanced.html#eliminating-quadratic-forms .
I believe this is what @Erling was getting at. I suppose I should feel guilty for not pointing this out (but thank Erling for doing so), just as I should feel guilty for thinking about, but being too lazy to point out earlier in the day of my post in this thread, that a QCQP would be better reformulated as an SOCP for the same reason as discussed in this thread. But I post more than enough, so I don’t feel guilty.