I think it's fair to say that until always indicates the end of a period of time, but:
It will happen until ... is specifying the end of a 'positive' period of time, i.e. the event will continue until the specified date (or period), and presumably then finish.
Hence a positive until is indicating the end of the event in referred to.
E.g. The season continues until the end of September means The season finishes on or about 30th September.
Analogous to Edwin's example: I swam regularly until I went to university means I stopped swimming regularly when I went to university: it didn't happen while I was at university.
It will not happen until ... is specifying the end of a 'negative' period of time, i.e. the event will continue to 'not happen' until the specified date (or period), and presumably then start happening.
So a negative until is in fact indicating the start of the event referred to, i.e. the start of the positive period.
E.g. The season doesn't start until the beginning of June means The season starts on or about 1st June.
To borrow Edwin's example: I had never played squash until I went to university means I started playing squash while at university: it did happen while I was at university.
Whether the period in question is inclusive or exclusive of the mentioned date is often ambiguous, but may be apparent from the context. I offer the following as 'general guidance' (but certainly not to be relied upon when the actual date is critical to you):
A positive until normally includes the specified date (or period).
E.g. The show is on until 28th June would usually mean that 28th June is the last night on which you can see the show.
Thus, the 'positive' period includes the date.
A negative until normally excludes the specified date (or period).
E.g. The show is not on until 28th June would usually mean that 28th June is the first night on which you can see the show.
Thus, the 'negative' period excludes the date, and hence the immediately following 'positive' period necessarily includes the date.
In both cases, the event you are actually interested in, namely "the show" is inclusive of both dates.
As Edwin has indicated, where the time of day is more critical, as in He is / isn't here until ..., there is an ambiguity as to the precise time, but normally both dates would be inclusive of the period that he is here.
He is not here until the 11th = He arrives on the 11th.
He is here until the 15th = He departs on the 15th.
Thus, his period here includes (at least part of) both days.