Oh wow Medicare? That seems a stretch even for USCIS!
If your flight is on the 27th, you can make an appointment for the 22nd, 23rd, or 26th since these are all 3 business days before your flight. If I were you I would set my alarm for this Friday 7/9 at 4:55AM so you are ready to make an appointment for Thursday the 22nd, or get up on Saturday 7/10 to schedule an appointment for Friday the 23rd. Technically you can make an appointment for the 27th if your flight leaves late at night, but that would be cutting it too close for me! Good luck and do keep us posted.