How many days hiking can the GPSMAP 67 store without syncing?

To learn about tracking and recording, you would be best to check out the Garmin support pages for the 67i,  (Track settings explained on the GPSMAP 66i/67i/86i/86sci.) How you setup the device will influence this. Basically you can record 20,000 track points, 250 gpx tracks, 300 fit tracks (activities). So how many per day depends on how often you record track points and the length of time you have the recording turned on for. How often you save them to your phone, whether expedition mode is on or off. There are some variables, so you are best to read the above article. Your phone does not need internet to sync with the Garmin explore, messenger or connect apps. It uses Bluetooth.