Can a person refer to themselves as Dr after completing the coursework for juris doctor degree?

No. A Juris Doctor (J.D.) is a professional degree, not a doctoral degree (Ph.D., Ed.D., etc.). While it's a demanding program of study, it doesn't confer the right to use the title "Dr." Only after earning a doctoral degree is it appropriate to use the title "Dr." A J.D. graduate may use "Esq." (Esquire) after their name, but not "Dr."