Organic merely tells you that the wheat that was grown, was grown in a more environmentally friendly way, ie. without the aid of pesticides. That on its own will have no effect on noodle stability.
They may be poor quality noodles, have gluten level issues, are they also wholewheat, do your regular udon have some form of strengthening additive like with ramen? Many noodles have kansui added, or whatever passes for that chemically (sodium carbonate) added to strengthen the dough. Of course it may not have been kneaded enough, too old, too dry.
I hate to say it, but I'd be willing to bet that gluten-free udon is about as practical as gluten-free seitan - the gluten is precisely what gives udon noodles the texture that makes them so special.
Having said that, I've bought frozen udon noodles that had tapioca starch as an ingredient in addition to wheat flour, and those were some of the best udon noodles I've eaten.
The other catch is that making udon noodles at home is hard - traditional recipes call for putting the very tough, springy dough in a plastic bag and stomping on it for a few minutes, as the only practical way of kneading it.
This site describes how to make udon noodles with tapioca - you could try replacing its flour with the gluten-free blend described here. I would also consider adding xanthan gum. If you're feeling adventurous, This e-book (not free, but looks pretty solid) discusses a wide variety of gluten-free flours and their properties, and might help you choose flours to use with the properties you want.
If gluten-free udon turns out to be impractical, I recommend soba noodles as an alternative. Those can much more easily be made gluten-free (although most commercially made ones available in the US do have wheat simply because it's cheap and bland) and can usually be used in the same dishes as udon. They have their own distinct texture, very different from udon, and a deliciously hearty flavor.
Best Answer
Many of the dried noodles that are marketed as "udon"—at least in my experience in the USA—are actually mislabeled, thinner noodles like Hiyamugi or even Sōmen. I would suggest buying the semi-dried variety that are usually packaged in vacuum sealed plastic. This variety is shelf stable, but it can often be found in the refrigerated section of Asian mega-marts. The noodles will already be almost full size before cooking.