I'm at a quarter system school and waited until I got my spring grades (actually even longer...) before I submitted my primary, as recommended by my prehealth advising office, and saw a very marginal increase in GPA (+.02) by doing so.
June 21 will still most likely be totally fine, but if you think you are a marginal candidate, then applying early will be more beneficial than the marginal GPA boost. If you really want to submit your spring grades later, you can always send them in an update, as mentioned above.