Punica granitum "Nana" is a dwarf pomegranate often used for Bonsai cultivation. It does flower but does not generally form fruit at least not any that would be edible. I have one in my garden at least 10 years that has grown to about 6 feet. Al

I've got a seedling that popped up under one of the old 'Wonderful' trees back home in AL; dug it up and potted it - it's spent the last 3 or 4 winters in my office, by the window. I've not yet brought it in for the winter, this year - it's probably leafless and near-dormant now; will re-leaf and bloom, but it never sets fruit, even when I tried hand-pollenating - flowers just drop off. It usually gets dried out and defoliates a couple of times during the winter.
I can usually keep it alive, but it typically looks pretty sad by the time spring comes.

I often walk through a property that is composed of 20 acres of a series of gardens. I came upon a tree with white pomagranates hanging all over it. I picked a few and they have pink kernals with almost no seed, very sweet. I would like to take a hardwood cutting and start another tree? Is this the right time of year to do this? And what type of pomegranate is this?

bonjo,
I'd wait until late Feb/early Mar, take several cuttings - pencil-diameter & 6-8 inches long. Wrap in damp(not wet!) paper toweling, place in a ziploc bag, and keep in the refrigerator 'til spring. Dip basal ends(It helps to cut one end straight across and the other at a slant - and remember which is which! - to remind you which end is 'up') in your choice of rooting hormone, stick in a well-drained soilless potting mix, and keep watered. In my experience, success rates run close to 100%.
You can also take semi-hardwood cuttings in late June/July and they'll root relatively easily, as well.

As to what variety it is(if it is, indeed, a named selection), I don't know, unless it's "Pink Ice"/"Sin Pepe" - but you might find its description in the book, The Incredible Pomegranate.

Here is a link that might be useful: The Incredible Pomegranate

Bookdiva:

You may be able to keep a pomegranate tree alive in greenhouse conditions or with interior plant lights in your climate zone, but I don't think fruit will be in the picture. I had dwarf Nana -- I think the same one Al describes -- for several years in my attached greenhouse, but it finally just pooped out in spite of reasonably good care and no temps approaching freezing. Just keeping a pomegranate tree alive from year to year might be a pretty good challenge in zone 5 unless you are experienced with tropical and sub-tropical plants.

Don Yellman, Great Falls, VA