I guess it's hard to give an answer to an ambiguous input that will please everyone. Personally I would have expected the 1st of the month.
It seems unusual that "next June" and "last June" both pick the 4th of the month (today's date); but just "June" gives the 1st of the month.
Also "next June" gives "Friday, June 04, 2010 12:00:00 AM" so it picks the day of the month based on today; but the time of day doesn't depend on the time now.
It seems unusual that "next June" and "last June" both pick the 4th of the month (today's date); but just "June" gives the 1st of the month.
Also "next June" gives "Friday, June 04, 2010 12:00:00 AM" so it picks the day of the month based on today; but the time of day doesn't depend on the time now.