If you are not a member of a union or under an employment contract you are considered an at will employee. At wil employees can be fired, have their hours changed, disciplined, etc for any reason EXCEPT discrimination. Discrimination is only on the basis of age (over 40), sex, race, nationality, color, religion, disability, marital or military status -- Federal EEOC. Your state may have other discrimination bases.
An employer can set out any rule they want as long as it is not discriminatory or is not applied in a discriminatory basis. Yes, an employer can prevent an employee from taking any second job as long as that rule is equally applied to all employees. Prior to looking for another job you should have looked at an employee handbook, asked HR, or looked at the company policies. This is a very common restriction.
The response from your boss about putting in your 2 weeks notice, indicates that she either felt you were going to take that 2nd job or that you already had taken it. You may want to go back to her and tell her that since you found out 2nd jobs were forbidden, you dropped it. If that is what you want to do.
As for the issue of your schedule having less hours then a part time job, it is within their rights as long as you are an at will employee without a contract that specifies the total number of hours you will work.