If your watch ended at midnight on the end of lets say the 8th of March. The time would be written in the log book as 2400 on the page for 08MAR07. You would mark through the rest of the unused space of the page with a single diagonal line and write "No further entries this page." (If I'm remembering this correctly, it's been awhile.) Your relief would start a new page for 09MAR07 and assume the watch at 0000. (Of course, this poor bastard is now standing the dreaded "Balls-to-four" watch where you might get 60 to 90 minutes of sleep before reveille.)
When we stood an underway watch with a round sheet. It would be marked with hours for 0000 through 2400. Technically, at midnight, the rounds would be taken twice, once by you and once by your relief, but one would be for 2400 on the day that just ended and the other at 0000 at the start of the new day.
Long story, short: At least for me when I was in the Navy
, the day started at 0000 and ended at 2400. The next day started at 0000; two events that occur at the exact same instant, but occurring on separate days.
Jonney