No, see, how it would work without timezones is:

• Everyone would use UTC and a 24-hour clock rather than AM/PM.
• If that means you eat breakfast at 1400 hours and go to bed around 400 hours and that the sun is directly overhead at 1700 hours (or something more random like 1737), fine. (Better than fine, actually!)
• Every area keeps track of what time of day daily events (like meals, when school starts or lets out, etc) happen. Though I think generally rounding to the nearest whole hour or, maybe in some cases, half hour makes the most sense. (And it's not even like everyone in the same area keeps the same schedule as it is now.)
• You still call the period before when the sun is directly overhead "morning" and the period after "afternoon" and similarly with "evening", "night", "dawn", "noon", "midnight" etc.
• One caveat is that with this approach, the day-of-the-month change (when we switch from the 29th of the month to the 30th, for instance) happens at different times of the day (like, in the above example it would be close to 1900 hours) for different people. Oh well. People will get used to it. But I think it still makes the most sense to decide that the days of the week ("Monday", "Tuesday", etc) last from whatever time "midnight" is locally to the following midnight, again probably rounding to the nearest whole hour. (Now, you might be thinking "yeah, but that's just timezones again. But consider those timezones. The way you'd figure out what day of the week it was would involve taking the longitude and rounding. Much simpler than having to keep a whole-ass database of all the data about all the different timezones. And it would only come into play when having to decide when the day of the week changes over.)
• Though, one more caveat. If you do that, then there has to be a longitudinal line where it's always a different day of the week on one side than it is just on the other side. But that's already the case today, so not really a drawback relative to what we have today.