Well to begin with, pines do not take very well from cuttings (they are almost always propagated commercially from seed or by grafting) so your chances of success are limited.

Second, allowing the root mass - if there is any - to freeze is not a great idea. Roots of even very hardy plants are the parts most susceptible to cold damage. Typically they are protected by the insulating properties of the surrounding soil. Anything in a container and especially in a very small container like a take-out cup will be extremely vulnerable.

I've attached a link to a previous thread from the Conifers forum that discussed growing conifers from cuttings. There is some excellent info there as to the correct timing and methods involved that will hopefully be helpful to you.

Unfortunately, I doubt your current crop of cuttings will take given the methods you've outlined. And again, pines are extremely tricky to propagate via this method and most hobby growers don't even try. OTOH, starting them from seeds generally produces decent, if slow, results.

Here is a link that might be useful: rooting conifer cuttings