As Factor30 has pointed out, March is actually a town in Cambridgeshire, so the statement "June was born in March" is referring to the fact that June was born in the town of March (nothing to do with the month of March).
The statement, "May was born in June", tells us that both twins were (probably) born in the month of June.
This sounds like a descendant of the old riddle - twin boys, born in March, who celebrated their birthday in September and when they grew up they married each other - answer was born in the town of March in the month of September and as adults both became Church of England vicars and officiated at their twin's marriage ceremony.