The same.

You're right, @BobBowles1. 身 means body (身体 being the bisyllabic construction). 健康 is health, as in the condition of your body, but also the abstract concept of health. In modern times, the concept of 健康 is expanded to include things like one's relationship to society. As such, you see the term come up in government propaganda quite a bit in relation to a healthy society, or healthy thoughts, or healthy attitudes. (I'm uncertain, but it seems like these useages might have been adopted directly from the west. I haven't done the historical research deeply enough to know for sure on this one.)

So the two are used interchangeably in the following circumstances:

1. When your body is synonymous with your health. (For example: Take care of your body / Take care of your health.)
2. And when it is not possible confusion will arise about the meaning of one of the words. (For example: He has a great body. / He has great health. Have very different meanings...)

Worth noting, if you are describing someone's physique that is 身材, although that one is almost always used in an aesthetic sense, in the way that it would in the example for No.2 above.

My understanding is that 身体 refers more to the body, and 健康 refers more to health. They are both related and at least in some contexts can be used interchangeably, but, for example, you can say 我的身体 meaning 'my body', whereas (I think) 我的健康 means more specifically 'my health'.

Maybe someone can correct me if I am wrong.

Well, I'm not sure it is wrong. They're still adding in a lot of other options for sentences. But it is also not how I've usually heard it said in China. Often times in Chinese it is more natural to leave the person addressed by the statement implied by context. After all, I can tell when someone is talking to me, or when a statment is addressed to the general public "you."

In fact, it is very common to see signs or hear people say "Please pay attention to health!" in Chinglish. I personally would translate this more artfully as "Take care of yourself." or just "Take care."

This is also very common when it comes to 我. My Chinese friends always tease me because they say Americans put "I" in every sentence, like we think we are the most important people in the world. In reality, it's just that English doesn't let you leave an implied subject like Chinese does. So many English sentences that start with "I" can be translated into Chinese without bothering to say 我 and in fact, sometimes it sounds very strange, especially if you have several sentences in a row, all referencing yourself.