No, you do not have to use your insurance. However, if you don't use your insurance you may pay significantly more for your treatments, including medications if you need them, although you can always try to negotiate a different fee. Sometimes medications (usually generics) are actually cheaper if you do not use your insurance to pay for them (otherwise known as "paying out-of-pocket"). Please do not use internet pharmacies that are not a part of your insurance; almost all of these are not legitimate. FDA testing of medications from such "pharmacies" shows that upward of 90% do not provide what you think you are buying, and in some cases, what is in the pills is actually toxic to humans.
If it is important to you to have absolute control over who knows about your private health information, you may want to consider not using your insurance.
Some providers of mental health services (otherwise know as behavioral health services) do not accept insurance, so you will be limiting your options if you are only willing to use your insurance.