She has not filed a green card application so she probably won't have an issue renewing her TN but if CBP or USCIS connects her to you, questions might be raised about her intent which could undermine her ability to renew her TN. So, yes, an H-4 EAD would be safer (unless the current administration kills this benefit).

We are married so are there steps we can take to decrease the chance of them "connect" her to me? Should we avoid crossing the border together?