June 4 is written as 6.4 and read as mu (6) shi (4). Mushi (虫) in Japanese means insect or worm or bug. So today is mushi no hi(虫の日), Day of Insect.
Bad tooth is mushiba (虫歯： literally insect tooth) in Japanese. Therefore Nihon shika ishi kai(日本歯科医師会), Japan Dental Association decided June 4 as mushiba yobo de (虫歯予防デー), Day of preventing bad teeth. It was practiced from 1928 to 1938. Since 1958 the week from June 4 to June 10 has become ha no eisei shukan(歯の衛生週間), Hygiene week for teeth.