Honestly I think the "you must apply June 6th (or whatever the first day is this year)" is a bit overrated.
The problem with submitting your primary later is that as June progresses, the processing time takes longer and longer. Ie, submit June 6th, you'll have your primary processed by mid-June. Submit July 6th, and you won't have it done until August 6th if you're lucky. By then, schools have released secondaries, and other people have started turning those in. You will end up submitting August/September secondaries, and that puts you at a bit of a disadvantage.
But that won't happen to you, because you can submit your primary for processing before your MCAT scores are out. If you submit your primary for processing in June, it should be done early, and you will just be waiting on that MCAT score to be complete at various schools. It should release in July, and especially if you pre-write some secondaries, you should be just fine. You'll be complete by July/August at the latest. You won't be waiting to be processed until that time.
That said, if you decide to wait til June, make sure you use that time to ace the MCAT. I'm serious. I have plenty of friends who delayed to "study more" and regretted it. They either slacked off, or studied too much and burnt out at the end. Make that time count, if you choose that option.