No, you don't need a degree before starting down the CISSP path, though a four year degree gives you 1 year of experience toward the required minimum 5 years of full-time work experience (in two or more of the eight domains of the CBK) needed for the CISSP. If you don't have that experience yet, you can become an associate of ISC2, until you've gathered that experience.
My suggestion is, if you're interested in Cybersecurity, just jump in and start learning in your areas of interest. When you're focusing on what you find fun and interesting, the rest tends to take care of itself.
Actually, you have up to 6 years in order to get the 5 years of experience.
Also, your experience can include years prior to taking the exam. The years of experience do not have to start at the time of the exam.