Why did butternut squash make my fingers dry and yellow?
It's definitely a very real reaction. I'm not sure exactly what in the squash it is that causes it, but since different people react differently - some people have strong reactions like you, some people have mild ones, and some have no problem at all - it seems to be some sort of mild allergic reaction. It's often called contact dermatitis, but that's a very generic term and doesn't really mean anything you don't already know. I'm not sure exactly what component of the squash it is that causes it, but I don't think it's carotene or potassium. It's likely some more complex molecule that tends to be present in various winter squashes.




Images from http://foodworld-eva.blogspot.com/2010/01/squash-reaction-on-skin.html.  

You can avoid it by completely wearing gloves while working with the squash, or if you're lucky, you have someone else with you who can handle the squash without problems. For many purposes, like soup, you can also simply cut the squash in half, roast it, then scoop out the good stuff. But if you do really need to peel it and cut into chunks, your best hope is just to try to avoid touching it too much. You'll also want to wash your hands immediately after, or even a few times while you're cutting if it takes a while.

If you do end up with this despite your best efforts, cortisone cream can help relieve the reaction afterward.

(And as for your soup, there's nothing wrong with letting more moisture escape in exchange for browning. You can always just add more water or stock back into the soup. This can even be a good thing, since it lets you replace water with a more flavorful liquid.)