NU is the pretty face of the death of private property in America. If you own something, you have control of it; zoning removes your control over property you own.
Standard zoning laws prohibit greeting clients and similar business activity business out of one's home. This is indeed a constraint on property rights.
But zoning laws existed for many years before New Urbanism. As far as I can tell, the New Urbanism advocates mixed-use zoning, which would actually give a land owner more options than previously.
For all I know, property right may be dieing in this country but I can hardly see this as evidence.
"Old" or "New" -Urbanism has never been about absolute property rights. City dwellers have, for generations, traded the conveniences of density over pure private property rights.
How would you feel about your next-door neighbor turning their home into a slaughterhouse or paper mill or adult bookstore?
Zoning is just one of many ways that people, through democratically elected governments, restrict private property rights in cities.
While Houston doesn't have formal zoning, a quick Google search (I've never been there) shows they do have extensive land-use regulation, including minimum lot size for single family homes (5000 sq feet) and requirements for parking for businesses.
So, if I wanted to place two single-family homes on my 9000 sq feet of land, or open a retail shop without providing parking, my private property rights would be infringed.
Yes, and I don't think anyone is hiding this. New Urbanism exists to turn areas which previously were only sparsely dense into much more populace areas. If you want your private property, you'll now have to move farther away from the current "big" cities to a smaller town or into an area where they still sell hundreds of acres of land for the price of a new york loft.