Can my Instagram account be set to private?

Yes, but this is not recommended at all while using an automation service like Suparise. You should keep your account public to maximize growth.

Suparise drives traffic to your Instagram using the Follow for Follow method. Your personal assistant follows users and likes posts to draw attention throughout notifications. Intrigued people will come to your profile and discover a private profile with nothing to see. Their interest will vanish.

Public is the way to go!