It's a little more complicated than just "Reporting". It's actually really simple.
1. Report for I-day. Process in.
2. 2nd day is when you take your "OATH". THAT is the magic date. If you get hurt or leave prior to taking the oath, then when you apply the next year, you start ALL OVER AGAIN. 100%. Do not pass go. Do not collect $200. Just like you never applied before.
3. Get hurt or have to leave AFTER TAKING YOUR OATH. You do NOT have to go through the application process again. Basically; as long as DODMRB clears you the next year, and whatever medically made you leave is all fixed; then you are automatically put back into the next year's class.
3a. If the injuring during BCT, after taking your oath, keeps you out of training for LESS than 8 days, you can be permitted to continue on with training and NOT be sent home. This 8 day requirement is waiverable. The class of 2013, this year's seniors, got hit with swine flu. Hundreds of basic trainees got sick. The 8 days were waivered and most were able to stay at the academy.
The good thing about the academy and backup plans is: The academy starts the LAST THURSDAY in June. Even if you made it 100% through Basic Training and then QUIT!!! That would only take you to the First Wednesday of August. You still have time to quit, go home, and start another college/university. Assuming you already had your backup plan figured out.