Which is correct and which is not?
- In June--July 1967, there were...
- Between June--July 1967, there were...
- Between June and July 1967, there were...
If I want to use "June--July 1967", how can I express?
You don't even say what you're trying to express. An approximate date? A few events within a given range? A set of inclusive dates?
For an approximate date, I would use around/or:
Around June or July of 1967, the storm hit.
For a few events within a range, I might use during/and (or perhaps between/and):
During June and July of 1967, it only rained five times.
Between June and July of 1967, it only rained five times.
To allude to the entire range of dates, I would suggest in/and:
In June and July of 1967, the weather was exceptionally hot.
For a longer range, I would use from/through:
From June through September of 1967, there was a severe drought.
Generally speaking, I would avoid using a dash. (Again, though, you haven't given us much to analyze). There are some contexts – like tombstones and journals – where dashes are appropriate.
Though the context is not clear, I would recommend referring to the dictionary meaning of Between. It fits in properly, if I assume your sentence is about a range or a shared/common aspect during June-July 1967.
"BETWEEN is used to indicate a range or something that is shared or combined."
And again, why not "during".
June-July: To describe sequences of dates/weeks/months, use of a hyphen (with no spaces between the hyphen and the characters) instead of the word “to” or “through” appears to me as an acceptable writing norm.