I managed to get it working with this query. Due mostly to the help of Lobo's answer.
SELECT *
FROM persons
WHERE DATE_ADD(STR_TO_DATE(birthday, '%m/%d/%Y'), INTERVAL YEAR(CURDATE())-YEAR(STR_TO_DATE(birthday, '%m/%d/%Y')) YEAR)
BETWEEN CURDATE() AND DATE_ADD(CURDATE(), INTERVAL 7 DAY);