Junior Associate (or Pre-Associate) role exists in only some offices in McKinsey. The role is designed as a transition role for Exp Hire (5-8 yrs exp) or Advanced Degree (PhD) or in some offices MBA (in India) before assuming an Associate role.
Junior Associate will almost do similar works like an Associate but with more guidance, as they have some solid prior experiences prior to McKinsey. A person usually takes that role between 6 months to 1.5 years before advancing to Associate.
Meanwhile, a BA is a role designed for a fresh or 1-2 yrs of exp undergraduate. BA in McKinsey is usually their first-time job, hence the guidance is a lot more than a JA, and they will need to be mentored more closely on day to day basis. BA usually lasts between 2-3 years before advancing to Associate or Senior BA.
Hope this helps!