He may accompany her in TD status and may study, but TD does not permit employment, including remote work from a U.S. address for a Canadian client. If he needs to work he must qualify for his own category.
TD allows presence and study, never employment
The spouse and unmarried children under 21 of a TN professional may be admitted in TD status for the same period as the principal. TD holders may attend school and, for children, this is often the main purpose, but the regulation is explicit that TD status carries no employment authorization. In the hypothetical, the husband runs a small carpentry business in Val-des-Monts; he cannot continue it from the U.S.
home or invoice Canadian customers for work performed while physically in the United States, because the prohibition is on working, not on who pays. His options are to remain in Canada with visits, to qualify in his own right for a category such as E-2, or to treat the U.S. period as a pause in his business.
Citizenship also matters for entry: a Canadian citizen spouse is visa-exempt for TD, while a permanent resident of another nationality must obtain a TD visa at a consulate first. Each family member is admitted on his or her own passport and relationship proof.