Naval Officer

You can join logistics branch in Indian Navy after MBA. You may refer to following link for further details. http://www.joinindiannavy.gov.in/officer1.php.
You are advised to appear for CDS examination conducted by UPSC to join Indian Army as an officer.

Indian Navy hires engineering graduates for a majority of their branches. Therefore, you may not be eligible to join Indian Navy as a BBA graduate. You may review attached link to assess your career options so that you can plan your education at PG level to join Indian Navy as an Officer: http://www.joinindiannavy.gov.in/officer1.php

JEE score is not used by admission in Merchant Navy colleges. If you want to join Merchant Navy, you are advised to appear for JAT conducted by Indian Maritime University. JEE score will be used by Indian Navy for selection in technical branch first time this year. Without historical data, it's difficult to predict what JEE score will be required for selection in Indian Navy in 2017.

Why not? Naval engineering by Indian Navy is one such option as it gives you the most reliable and inexpensive option. If not, then try the following link:
https://engineering.shiksha.com/be-marine-engineering-colleges-in-india-categorypage-2-62-1-0-0-1-1-2-none-1-0

You are requested to visit following link for details on joining Indian Navy after B.Tech ECE: http://www.joinindiannavy.gov.in/officer1.php