Making crisp hash browns

Would microwaving on a paper towel help remove ice crystals and help avoiding soggy potatoes?

I don't believe this would be a good approach. To microwave long enough to drive off moisture, you will wind up cooking the potatoes in the microwave. Then when you attempt to finish cooking, the interior of the potatoes will over-cook before you get the right degree of browning.

What is the best way to drive off moisture to help make them crispy.

First you need to make sure they are thawed. Per the other answer, rinsing in a colander under cold/lukewarm water will accomplish this. Of course, this will add moisture to what's already there, so it will be even more important then to remove the excess.

Once thawed, you can remove the excess water by wrapping the potatoes in a clean linen dish/hand towel, and then twisting the opposite ends of the towel to create pressure on the potatoes, which will squeeze the water out of them.

Then once you've gotten the water out, cook them as usual. Make sure you use sufficient oil or butter to ensure a nice browning rather than just drying out and toasting the potatoes.

Bonus tip: when I make hash browns, I use fresh potatoes and I find that the potatoes start to oxidize once shredded. This can result in a slight green tinge to the cooked potatoes, which I'm not a big fan of (it doesn't affect the flavor at all, but I prefer the fresh color). This can be avoided by tossing the shredded potatoes in a dilute vinegar solution; I find that an 8:1 solution, e.g. 1 cup water to 1 oz. vinegar is strong enough to prevent the browning while not imparting a vinegar taste to the potatoes. Naturally, do this before you squeeze the water out of them. Of course, if you like a little tangy flavor in your food, you might try a stronger solution, as well as experiment with other types of vinegar or acids (e.g. lemon juice). :)