There are two distinct calendar issues involved here:
Different calendars - eg in 1640, the Julian calendar was used in England and the Gregorian in France, so that 28 January in England was 8 February in France (but the same day of the week) - ten or eleven days difference depending on the year This mostly shifted the exact date, and only changed the year for some dates in late December/early January.
Different definitions of the year end - sometimes the year was counted as changing over on 1 January (the modern convention), and sometimes it was counted as changing over on another date (in England, this was 25 March).
The two calendar changes were often connected, but not always. (The Wikipedia articles on dual dating and old/new style dates are helpful here, as is this page from Medieval Genealogy)
There are a couple of nice contemporary examples from the Folger which show how people tried to represent this - the first gives the day of the month in both Gregorian and Julian calendars, as "29/19 Jan" (it was written in Paris to be sent to England), and also gives the year as "1650 [new style]" to indicate it is 1650 by the modern method, rather than 1649 by the old method.
The second example doesn't do anything about the date (it is just "March ye 6") but splits the year as 1736/7. This writer was not concerned with Julian vs Gregorian, as they assumed everyone reading the document would be working in Julian, but they still had to worry about the ambiguity of the year.
Your example is closer to the second situation. The contemporary Julian calendar gave the date as 28 January, and our English letter-writer would have reliably assumed that everyone in England would agree it was 28 January. They would not have worried about a Gregorian date unless they specifically needed to - eg they were corresponding with someone on the Continent.
They would, however, have worried that their correspondent might not agree on what year it was, even within the same country. Under the "new style" of counting from New Year's Day, this date is in the new year of 1640; under the "old style" of counting, it was still in the old year of 1639. If you were quite traditional and wrote "28 January 1639", your reader might interpret it using a new-style date and think the document was a year older than it really was - under that system, January 1639 was twelve months ago.
(For a modern comparison, think about financial years or tax years - "...when the 2020 financial year ends in April 2021...". It's a bit confusing, certainly, but also generally accepted.)
So the convention was to write 1639/40, to avoid any ambiguity, or to indicate it in some other way. (I can't work out how to show the right markup here, but you often see 39 over 40, with a horizontal bar not an oblique slash; the first Folger example uses this for the date).
So, what is the conventional way of representing this now? A lot of publications silently convert to "modern" years, so would give your date without any qualification as 28 January 1640 - using the Julian calendar as in place at the time, but the modern year-dating convention. This is used by eg the Oxford Dictionary of National Biography. Alternatively, this is from the Modern Humanities Research Association style guide:
If it is necessary to refer to a date in both Old and New Styles, the form ‘11/21 July 1605’ should be used. For dates dependent upon the time of beginning the new year, the form ‘21 January 1564/5’ should be used.
Your author is using the second approach suggested here - writing 28 January 1639/40, splitting the year because it could be interpreted both ways - it lets a reader know for sure what is going on, rather than trusting they've all been standardised correctly. However, they're not splitting the day of the month because in the context of 1640s England, all dates can be assumed to be Julian; adding a Gregorian conversion would be anachronistic and potentially just confuse the reader.
Finally, the notation - I can't seem to find definite references to this in many other style guides, but from personal observation, the slash is much more common than a hyphen - as you say, a hyphen might be misread as a date range. I would definitely recommend this as the punctuation unless, eg, you are wanting to make an exact transcription.
