As far as I can tell, the Federal Mailbox Restriction Act only addresses "mailable items", and seems mostly concerned with preventing the private delivery of mail (into mailboxes) that would otherwise have gone through the Postal Service.
If you wanted to be a real dick about it, you could probably complain to someone and, at best, your neighbors might be asked to refrain from using your mailbox to deliver notes. Not to mention that your neighbors are probably capable of making life much worse for you, than you are for all of them.
Seems to me though that they made a perfectly reasonable request in a polite and civil manner. Just show a bit of courtesy like Knox Joe said, and wait till you're a few blocks away before getting on it at 6:30am.