Well, depends on what you mean by "proper." A direct transcription of her first name in Japanese using the most common system we use here (Hepburn) is Jun. So that's proper in that it's regular, but it's also not because that's not how she writes her name in English any more. She's been using "June" since 2005 or so, and it's on both her English and Japanese profiles.
"June" could be the Japanese ju-ne (じゅね), but I'm fairly certain it's meant to be the English name "June," which is pronounced more or less the same as the Japanese name Jun ("June" would usually be written with a long vowel, but I'll give it to her). So it's not really made-up or anything terribly drastic, but just a westernization of her name. It's not a one-to-one mechanical substitution of the characters, but since it's pronounced the same and she uses it reasonably consistently, I don't think it's any less proper than Jun when referring to her in English.
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