I've propagated Walnut trees, and documented it in my blog, with photos.
Your photo indicates a regular Persian Walnut tree.
Squirrels are notorious for cleaning off nut trees, but they also forget where they hide all their nuts, so I am surprised you don't have at least a few young trees coming up every Spring.
Walnuts have deep taproots, so they don't like being transplanted.
If you do manage to collect a few walnuts before the squirrels get to them, they need to go through a period of cold temperature stratification. Basically, they need to go through a period of cold temperature for about 100 days or they won't germinate. The safest place to do this is in your refrigerator. Keep them slightly humid. In the Spring, take them out and plant them in pots.
They will still need to be protected against squirrels for at least the first year. Even after the nut germinates, if squirrels smell them, they will dig up the nut and take them. Even if the tree is four months old and 1 foot tall, squirrels will nip off the tree at the base and dig up the nut, killing the tree for the sake of that nut. Squirrels are pesky critters. One way I found to protect the trees from squirrels was to wrap the tree in a
chicken wire teepee. The trees were in five gallons planter pots and each one got a cone of
chicken wire wrapped around it.
That was the only way that worked for me.
After the first year, the chicken wire can come off the following Spring.
I think, by this time, the smell of the nut has disappeared and the squirrels ignore the young trees.
Regarding trying to clone the tree by taking cuttings, I don't think it can be done.
If you can successfully do it, that would be wonderful. Post back and let us know.
But if you want a clone of that tree, a better option would be to graft scions from that tree onto rootstock of another Persian Walnut tree, or a Black Walnut tree.