Under the lang options provide an option to specify the length of the month abbreviation.
French abbreviate their month names to 4 letters not 3. With only three June and July both abbreviate to "Jui". I think the easiest solution would be to provide an option that we can set under "lang" that specifies how a long a substring to take when abbreviating month names.
An alternative would be the option to specify an array of abbreviated names for weekdays and months. I have submitted a request for this, since I need it because some abbreviated names in dutch aren't really abbreviations i.e. the month march (maart in dutch) must be abbreviated as 'mrt' not 'maa'. However as this would also solve your issue, perhaps I can enlist your support for my request? see: http://highcharts.uservoice.com/forums/55896-general/suggestions/2296431-abbreviated-months-and-weekdays?ref=title