As Big Jim said, go for it -- learning by doing is a great way to learn things ... what does and doesn't work. If you don't mind the expense of some failures along the way, then try it and see. There are some things that you will learn about green wood such as it is very easy to turn, but is also fragile and can easily tear or chip out. Of course, the term "green wood" can cover anything from sopping wet cut in early spring to almost dry, but still a bit of moisture.
Moving and cracking go hand-in-hand. When something restricts the wood from moving, the result is generally splitting. My personal recommendation is that for pen turning it is best to cut the wood into reasonable size sticks and let them dry before turning. It won't take too long although making pens by Christmas would be pushing things if the wood is really wet. Bottle stoppers and handles on the other hand are not a big problem. A little shrinking and warping can be tolerated in most cases although I would prefer to use dry wood.
As a final thought, I would add that the wood on pen tubes is so thin that it will most likely be dry by the time that you finish turning them. Because green wood is more fragile than dry wood, it would be best to use a skew where you can make slicing cuts rather than something like a carbide scraper tool that tends to tear the grain. Sanding will be the most risky part of this because a lot of heat can be generated which can lead to cracking. I would recommend that you slow the lathe way down and use very light pressure with the sandpaper. If you feel the sandpaper getting hot, you will know that the wood itself is even hotter ... and it is time to get some fresh sandpaper and lighten up with the pressure.
While experienced turners often try (perhaps too much) to save new turners from having to learn which things do and don't work the hard way, I believe that a better approach to learning might be sharing knowledge while encouraging experimentation. By experiencing some of the basics on wood characteristics I believe helps develop one's intuition about the behavior of wood which is knowledge that will serve you well in the future.