I think this means "Do you work 24-7?" "Are you workaholic?"
I thought of this possibility, too. よく働きますか could mean
Do you work a lot. よく could mean
often,
a lot, or another depending on the context. Here is the link to
大辞林. When I see the phrase
よく働く alone, the phrase
a lot basically comes up on my mind. In addition, because of the combination of よく and 働く, it further morphs into
hard.
Do you work hard was the first possibility I thought of specifically for the original question in post #1.
However, when I see the question よく働きますか on the whole but without any context given, personal debating starts to happen:
Does this person really mean Do you work hard? It's very unusual to ask this type of question to start the conversation. When we decide to use the よく ... しますか construction to start the conversation, we usually say 海にはよく行きますか, 釣りにはよく行きますか, あなたはお子さんとはよく遊びますか, etc, where all of these よく mean often. Doesn't the original question mean Do you often work?
Then,
Do you work hard echoed in a more unusual way than
Do you often work given no context, which is why I came up with posts #2 and #4.
Because of such unusualness, Wishfull and I decided to land on different interpretations first. And Wishfull reached an unusual example conversation which ordinary people could have:
A: Do you often work?
B: No, I often take off.
And I reached an ordinary conversation which unusual people (millionaires) could have:
A: Do you often work?
B: Not often. Three weeks a year.
In principle, we seem to be in agreement on what よく働きますか could mean given no other context than the situation where it's on the beach.