Aprender Inglês Faz Bem Para a Saúde?

Sim.

"O aprendizado de novas línguas ativa neurônios e exercita a memória, tornando a pessoa capaz de processar as informações em mais jeitos diferentes. Isso retarda doenças degenerativas, como o Alzheimer, e melhora seu desempenho em atividades intelectuais em geral, provas e trabalhos em qualquer área do conhecimento." (correiobraziliense)

Qualquer coisa que estimule seu cérebro a trabalhar te fará muito bem

There is that imagery the scientist used, that the brain connections like the trails in a forest, the more you walk back and forth in that trail the easier it gets to you learn and recall the information.
That will happen with any kind of learning, but as we are in a site that deals with English I daresay we have plenty of material to make associations (hence connections as well).
We know for a fact that music can be used as a treatment to soften the effects of Alzheimer's patients, languages certainly will be another way to 'jog' our memories.
Of course it's not only that, there are other factors like healthy eating, exercise, etc. The brain and the body (for that matter) is to be seen in the context of an integrated system.
Some time ago; for instance, people would talk of old people - they shouldn't work, they shouldn't do this or that).
Now we know that the age isn't exactly chronological, there is this concept of biological age, same goes
To the brain works.

Once we have take that in account, the answer is "yes", learning something (including English here) is good to our health, brains and wellness. And it sometimes helps to engage in a conversation with some English-speaking person, to get information in another country, to laugh at some joke, read the news, and so on and on.

Ref. kids.frontiersin