I'm a newly commissioned Navy Nurse and I think I may be able to help. I have experience with NCP, Direct Accession (DA), and ROTC.
It is up to you but if you join the Navy now without a Bachelor's degree you'll be enlisted, not an officer. You may have been talking to an enlisted recruiter instead of a medical officer recruiter, which makes a HUGE difference. My suggestion is to finish school first and then join the Navy, or apply for the NCP program. Don't stop school while you're in the middle of it to go join the Navy. You'll be taking a 4 year break from finishing your degree and it is no guarantee you'll be working in the medical sector.
You can only apply for the NCP program in your Junior year, so it is a good idea to start getting your application ready now. It can take a while to get the items together. Right now the NCP program is offering a 10k stipend for your Junior and Senior year with a 3 year commitment after graduation. DA (joining the Navy Nurse Corps after graduating) is offering 20k for 3 years or 30k for 4 years. The only real difference between the two avenues to get into the Navy Nurse Corps is how much "time in service" you will have. If you do NCP the time you were in school as a NC counts towards to your "time in service" It is possible for someone who did NCP to have 2 years of "time in service" opposed to someone who does DA.
Also, if your school has an ROTC program that is another route you can take to get into the Navy. Yes ROTC is a little more intense and there are more requirements, but you have to remember the Navy is decreasing manpower. Right now the Navy has one or two boards to review packets at the beginning of each fiscal year (this has been true for both NCP and DA the last 3 years) so there is a small window to get your application in.
The Navy Nurse Corps is one of the most competitive programs the Navy has so having a high GPA, good recommendations, and an impressive resume is important. Let me know if you have any more questions, I am more than happy to help :)