You can technically apply for a green card but border guards can and do use it as evidence that you don’t intend to stay “temporarily” and deny your TN as a result.
They can reevaluate your TN status at every crossing, and there are many stories of this happening to people returning from holiday when I was at microsoft. The TN is efficient to get but has its downsides.
That's correct so when pursuing a green card while in TN status, the applicant should only travel internationally after commencing the green card process on an advance parole travel document (not on a TN) or get his or her green card through a process abroad, called immigrant visa processing. An applicant for a green card in TN status should never seek admission in TN status if he or she is the beneficiary of a pending or approved I-140 petition unless he or she is immigrant visa processing and only then after consulting with an immigration attorney to understand the issues, process, and risks of doing so.
They can reevaluate your TN status at every crossing, and there are many stories of this happening to people returning from holiday when I was at microsoft. The TN is efficient to get but has its downsides.