Cucumbers can be cloned. There would be two ways to accomplish it. One is through tissue culture, expensive and not likely to be as reliable as layering, cuttings, or from seed. Embryo harvesting is tedious and requires technical skill and the tools to accomplish it.

The layer method would by to "brush" rotting hormone onto a node and then cover the node with soil. Cuttings would not be very reliable since annuals such as cucurbits are prone to rot and are short lived. The same issues are present and relevant for layers but it is still possible.

As for the reasons you are interested. When "garden plants" germinate from the seed of plants that were planted the previous year they almost always have reverted to the traits of one or a combination less desirable of its parent plants. For example your cucumber, it has large leaves and grows rapidly. How many flowers have you observed and has it set fruit?
My guess is that you have seen few if any flowers. Hence, one of the parent plants had the same attributes and was a poor fruit producer. The other parent was much slower growing and had smaller leaves. But, it flowered prolifically and produced moderately sized fruit that was tasty but, had lower sugar content because of retarded growth and smaller leaves. A cross of the two was made and the results produced a plant with medium sized leaves enough flowers to produce a sizable crop that was above average in size and sweet tasting.

The results of such crosses is called Hybrid Vigor, this is what plant propagators are trying to attain when they breed in this manner.

I stated the above not to discourage you but to keep you from being disappointed if things do not "turn out" with your volunteer plant.

Thanks for your help. The plant has tons of flowers. The seeds I use were all heirloom and I believe that the particular seed that plant came from just had a beneficial genetic mutation.

So you really think there is next to 0% chance of using a cutting? I want to find a way to make lots of this plant type.

If I used the seeds from this plant then it would be alot less reliable since each seed would be slightly different right?

No problem, since it was an heirloom it may produce seed that is usable. Heirlooms are a little different in that they are specifically bred to produce fruit and reliable seed. It is one of the reasons that heirloom vegetables are gaining popularity.

The chances of making cuttings are small. There are physiological constraints that come into play. Again, this is an annual plant and they have very soft tissue making it more prohibitive.

If the plant is the offspring of heirloom plants the chances are increased for getting good and reliable seed. There are too many factors to list here that will determine that.

If you have the time, patience and desire then it may be worth it.

The change in my stance is due to the addition of the heirloom progenitor(s) information.

Thanks for your time and explanations hortus. Right now I grow and sell to a local nursery. All of my plants are organic and heirloom. My seeds came from Seed Savers Exchange.

This is my first year attempting to clone plants though so I am a bit new at it. So are you telling me to just save the seeds from it and really not attempt to do cuttings?

I know all seeds are slightly different from the parent but since this plant is so vigorous does that mean that a good chunk of the seedlings will share most of the genetic traits of the parent plant and that I should collect the seeds and simply thin out whatever doesn't appear to have positive mutations next year?

By doing this will I continue to make my heirloom cucumber plants stronger and stronger each year? Thanks for all the advice.

For the cucumbers cuttings are not feasible. Without knowing a great deal more about the parents and the genetics I can not say how many. But, seed collection and then selection of seedlings is your best bet. Try doing some research on breeding plants and then pollination by hand for selected seed production. It is time consuming and tedious but yields are more reliable and you know what is what, if you will.

The selective pollination and selection of parents will yield more reliable. It gets very "deep" very quickly. If you are serious it can also get into some serious record keeping and a lot of work.

Letting "nature take its course" is unreliable and will yield far fewer positive results. Most of what you are seeing are not likely to be mutations as such. It is far more likely to be Dominant v. recessive genetics. It will require some reading and study to understand and to get "good" and manipulating the genes. Mendel would be a basic start.