Bought a house in Dayton, Ohio, and want to plant some white walnut trees. (Temperatures: I've seen it get down to like -5F, a few years ago, but looks like it rarely gets below 15F.) I've found a source of nuts that MIGHT be plantable (though I'm not sure how they've been keeping the seeds), but I'd likely need to stratify them, which would take a few months and then we'd be into summer. Can you still plant white walnuts in the summer, or would that mess with their growth somehow?
It depends on how do you plant it.
If you buy it in a pot, and you can plant without touching much the roots, it doesn't matter so much the period you plant it. But if you get with bare roots, fall/winter/spring (depending on your zone, the plant, and wild life around you) is way better.
But: all potted plants could suffer when planted outside. Usually trees are keep outside, so not much a temperature shock, but sun could be a problem if the tree was keep in shadow and you plant it in full sun on most sunny and warm days of summer. I would put the pots near the final location for one week, and if there is no sunburn (but check daily), I would plant them without much worries. Else I would move them to a half-shadow and then from time to time into a more sunny place until you can plant them.
My major worry is about transporting such plants. My experience tell me that branches are often damaged, and this is more difficult to recover.
In case you're not aware, the butternut tree is endangered in Ohio due to its high susceptibility to the non-native disease, butternut canker. Researchers are still searching for disease-resistant varieties, but at the moment there are none commercially available. If you plant nuts from a surviving butternut tree, you may end up finding such a variety and helping to save the species, which would be wonderful. But you should go into it knowing that the most likely outcome is that your butternut tree(s) will not survive for more than ten years.
Apparently free-standing trees are somewhat less susceptible to butternut canker, so you're more likely to have success if grow your butternuts in full sun, away from other trees. If you can get saplings or seeds that are hybrids between butternut and Japanese walnut, those are supposed to be resistant to butternut canker.
Black walnut, on the other hand, is quite resistant to this disease. This is why black walnuts are still quite common in native Ohio forests, while you very rarely see butternut trees.
A good place to get advice on how to grow butternuts from seed would be a native tree nursery that sells them. You might think they wouldn't give advice that would help you not buy their product, but many of these places are happy to help anyone who is interested in growing native plant species. Here's a list of Ohio native plant nurseries that grow trees. Of those, Riverside Native Trees and Scioto Gardens definitely grow butternut trees.
If your potential source of butternut seeds doesn't work out, you might be able to gather some in the wild. According to iNaturalist, there might be a butternut growing on public property in Taylorsville Metropark. It's not a confirmed ID, but the person who posted it is a naturalist and seems to have a pretty decent track record at ID-ing plants. The actual nuts of the butternut are quite large and relatively easy to spot on the ground (speaking from personal experience).