Data Analyst or Business Analyst experience to become a Data Engineer?

Rephrased question: Which job would lead to a data engineer job? Is it data analyst or business analyst?

Answer: Sorry, there are two things wrong in the question from my point of view

1. There is a wrong assumption in the question that to become a data engineer there is a need to first get into some other job.
2. A business analyst or a data analyst is a job of its own and is not an entry-level job to lead to data engineer job. From a BA the natural progression is that you become a senior BA, and similarly from a data analyst you become a senior data analyst and so on. 

Now that there is some clarity, from any of these roles if the person is interested, is ready to learn/unlearn and there is an opportunity in the organization the person can switch between these roles. As data analyst is in general more technical than a business analyst (in the sense of requirements analyst) it would be easier for a data analyst to switch to data engineer role compared to a business analyst. However, the difference is, these are horizontal moves and not vertical moves as implied (maybe unintentional) in the question.