The proper noun in your sentence is the word "June". The general rule is that one should capitalize when one writes about the days of the week, months of the year and holidays. Hence, it becomes "Since it was June, she wanted the party to be outdoors."
Also, there's a correction in the last sentence; the subject should always be followed by a predicate. For it to make sense, it has to be "She wanted the party to be outdoors in June."
