Much of "the woods" is owned by the US government, where your chances of any degree of success are highly variable. It is extremely unlikely that you can get away with it at all on a military base or in a national park. You may be able to get away with it for longer on Forest Service land (legally speaking, you're supposed to move along after 14 days), but if you're looking for a permanent legal claim to the land, that will not happen without an act of Congress.
If public domain land has valuable minerals which you exploit, you may be able to chop down trees and build a cabin, but until Congress lifts the moratorium on mining claims patents, you cannot gain title to the land. (Public domain land is land not set aside for a specific purpose, such as a national park or wilderness area).
Another possibility is to seize the land through adverse possession, as long as you satisfy the requirements for such an action in the state in question. Chopping down trees and building a cabin probably satisfy the requirements of actual possession, openness and notoriety. You would have to continuously live there for 5-30 years, depending on state, and have to have exclusive use of the land. If you get found and the owner tells you to leave (whether or not they get a court order), or if they say "I'll let you stay for a while", or they do a bit of landscaping, then you can't take the land (or, the clock restarts). There are a number of state-specific quirks such as whether you have to believe that the land is actually yours. Also, you can't dispossess a government.
At some point, you will have to deal with the county, since you built the cabin without a permit.