What is a non-tenure-track assistant professor position in the USA and is it harmful?
Actually, in this specific case I think it may just be a mistake in the ad.

The application page linked from the Chronicle ad just describes the position as "Assistant Professor" and doesn't say anything about tenure one way or the other.  Certainly the default for an assistant professor position would be that it is tenure track.

Also, I attended UC San Diego as a grad student (in a different department) and don't ever remember hearing about any non-tenure-track assistant professors.  Full-time teaching-oriented faculty at UCSD have the title  "Lecturer".

So I think there's a good chance that these are ordinary tenure-track assistant professor positions, and that someone just clicked the wrong box when submitting the Chronicle ad.  

I would suggest getting in touch with the department for clarification.