I do not have a menstrual cycle without taking birth control. Am I healthy without a cycle?

PCOS. See your doctor and ask if you may have polycystic ovarian syndrome. It is a common of what we call secondary amenorrhea (when you don't have periods).
Risks over time. If a female doesn't have a few periods/year, then she is may be more vulnerable to the effects of unopposed estrogen which could increase her risks for both endometrial and breast cancer.
Absent Period Risks. Going too long without a menstrual period can put you at an increased risk for uterine cancer. Please speak with your physician to regulate your cycles or utilize Progesterone to protect the uterine lining with regular withdrawal bleeding.
Healthy to have. Several studies have shown that various medical problems can arise/be associated with women who have irregular or infrequent menses including hyperinsulinemia, Insulin resistance, dyslipidemia, cardiovascular disease, endometrial hyprplasia and endometrial (uterine) cancer.