Youth employment law in NH is laid out in RSA 276-A.
New Hampshire allows the employment of minors under certain conditions. Minors aged 12 to 15 must obtain a New Hampshire Youth Employment Certificate, while minors aged 16 and 17 may work with a parental permission form. Minors aged 12 to 13 face extreme restrictions on employment prospects based on industry; minors aged 14 and 15 face heavy restrictions and are largely limited to the food and service industries. Minors aged 16 to 17 face fewer restrictions and may work in a greater set of industries, especially those who are not enrolled in school.
Also, if a minor is not working for their parents, grandparents, a guardian, or in newspaper delivery, they are likely to need to be at least 12 years old. If they are 12 years old, but under 16, they cannot start work earlier than 7 AM and can't leave work later than 9 PM. They also cannot work more than 3 hours on school days, and are capped at 23 hours per week during school weeks. On non-school days he may work 8 hours a day, and during school vacations, they can work 48 hours per week. There are exceptions made for farm labor, newspaper delivery, and working for family, as well as other exemptions and requirements to accommodate for school.
The full details of work time and industry restrictions may be found at the New Hampshire Department of Labor.
The New Hampshire government answers questions and provides RSA law citations for these restrictions on this page.