No. A TD spouse may accompany the TN professional and may study, but TD does not authorize employment. Dental hygienist is not an Appendix 2 profession, so he would need a separate category with its own requirements, plus any state licensing that applies to hygiene practice.
Plan the spouse's status on its own facts
The household should treat the hygienist's situation as a separate case. TD status is derived from the TN principal and gives no work permission, even part-time or remote work for a Canadian clinic performed from US soil is outside what TD allows. Because his occupation is not listed in Appendix 2, TN is not available to him in that role; a different category would require its own employer, petition and eligibility analysis.
State dental boards license hygienists separately, and a licence is a practice requirement rather than an immigration status. If the family's budget depends on two incomes from the first month, that assumption should be tested now. Options include the spouse continuing Canadian work while physically in Canada, pursuing study in TD status, or delaying the move until an independent route is identified.
Children in TD may attend school but face the same no-work rule.