There are multiple things to consider here. If you'll have access to seeds next year, or are willing to seek out seeds from someone that will have them then, I'd just wait and grow more, from seeds, next year. Plant the seeds where you want them to grow permanently. You'd likely have better luck and superior trees, later on.

If you do decide to plant these, you'll need to bare-root them and prepare their root systems before planting. You'll need to remove j-rooted taproots just above where they begin to be deformed. You'll also need to address any future girdling roots (potbound roots).

One possibly substantial downfall of having a modified root system is that buckeyes tend to be particularly finicky about dry soil/drought. Trees (and I'm specifically addressing buckeye trees, not necessarily other types of trees) with natural root systems (naturally developed taproots AND naturally developed secondary roots) can be significantly better adapted to dry periods/conditions.

And, yes, anytime you are moving a plant outside, hardening off should be part of the process. Without it, plants can suffer significant shock or even be burned to a crisp.

Here is a link that might be useful: Planting a Tree or Shrub

Thank-you for your input! I do plan to sow more buckeyes as I have a steady source for the seeds but THESE seedlings definitely have sentimental value to me and I'm not willing to give up on them that easily! My property is a bit odd...the land is very rocky but woody...so there's plenty of shade to be found. I will plant them out and create a tent around them to protect them from wind and sun until they have hardened off. I know they will probably die or be terribly stunted but I'd like to give them a chance!

I find most buckeyes to be pretty easy to grow from seed and forgiving of bad treatment. In my experience Aesculus pavia is one of the easiest. I grow a number of them every year. I collect the seeds in the fall just as the husks start to split. I just throw them in a ziploc baggie, label them and throw them in the back of the garage fridge and forget about them till spring.

I just dug out a bag 2 days ago and most of them had the root radicle coming out. I stick them a couple inches apart in a 6" deep crate full of ordinary potting mix, stick in the shade, and water periodically. About September, I just dump out the whole crate, select all the best seedlings, and repot in a 1 gallon pot using a coarse bark mix with a little compost.

I overwinter them outdoors in the containers, and then by the next fall they are ready to move up to 3 gallon pots or plant directly into the ground. I do give them a small handful of organic fertilizer every spring, but they don't get much pampering otherwise. My growing area is very shady, so I get away with watering the containers only about once a week.

I am no nurseryman, but this has worked well for me, so yours at least have a fighting chance.