Let W, H, S be the men who come once every two, three, and seven days respectively.
Because W visits every 2 days, his first visit of the month must be on the 1st or 2nd. Similarly, H's first visit must be on the 1st, 2nd, or 3rd. But each pair of Monday, Wednesday, Friday are separated by at least 2 days, so W must first visit on the 1st and H on the 3rd.
Clearly, the 1st of the month must be a Monday, Wednesday, or Friday.
If it's a Friday, then H's first visit is on a Sunday, contradiction.
If it's a Monday, then S's first visit is on Friday the 5th and the only other even-date Friday in the month (i.e. day when both W and S are in the gym) is Friday the 19th, on which day H isn't there. But we know all three are there on some day, so again, contradiction.
So the 1st must be a Wednesday and S first visits on Monday the 6th. Now the only odd-date Mondays in the month (i.e. days when both W and S are in the gym) are the 13th and the 27th. W is there on all multiple-of-3 dates, which includes the 27th but not the 13th.
So the answer is
the 27th of the month.
This isn't really a new puzzle. Monday, Wednesday, and Saturday are still three maximally separated days in the week (two 2-day gaps and one 3-day gap). So we can argue exactly as above (let Saturday be kMonday, Monday be kWednesday, Wednesday be kFriday, and insert "k"s everywhere in the above argument) to get the same answer as before:
the three men are all in the gym together only on the 27th.
The reason this is a good puzzle (and not just a maths problem) is because it's not obvious where to start. Obviously we could just brute-force it by considering all the possible days of the week on which the month could start and all the permutations for assigning W, H, S to Monday, Wednesday, Friday - but this would take a long time without a computer and wouldn't be very satisfying. The better approach is to examine the puzzle for a while until it becomes clear what given information is going to be most useful and how to deduce as much new information as possible in the fewest steps. In only the first few lines, my answer has already thrown out almost all of the possibilities a brute-force approach would consider - but it's not immediately obvious that that's the right way to start on the puzzle.