Kelley at Preparedness Pro published an article called Safely Preserving Eggs. She believes that the answer to your question is yes, the cartons should be gently flipped upside down about once a month. This is to preserve the integrity of the egg. She also seals them individually for the best protection and longevity.
This is her opinion:
There are several methods you can use and I’ve written of a couple of ways previously, but my favorite one (because it’s the easiest) is using mineral oil.
All you have to do is warm a quarter cup of mineral oil (just about 10 seconds in the microwave will do).
Set your eggs outside of the carton (because it will be hard to get them out one you start using the mineral oil). Put on some food handling gloves (I buy this at one of the warehouses). They are easy to use for safe food handling but they are also a lot less expensive than medical gloves and yet in some circumstances they can be used instead of medical gloves. They are not puncture resistant though, so exercise caution if using them for that purpose).
Dab a little bit of the warmed mineral oil on your hands and then pick up an egg. Run your oiled hands all over the eggs, making sure to cover it completely with the mineral oil. Don’t worry if you’re putting it on too thick or thin, just so long as every part of it is covered. When you’re finished with an egg, put it in the egg carton, small pointed side down. A quarter cup of mineral oil should easily do 4 to 6 dozen eggs.
Now, store your egg cartons in a cool, dry place. You want the temperature to be about 68 degrees for long-term storage—otherwise storing them like this in your regular room temperature is just fine for a few weeks. Remember, the eggs come out of a warm hen. I’m always asked if this will help the eggs keep longer in the refrigerator too. The answer is yes.
You’ll want to set a reminder on your calendar or cell phone to flip your eggs once a month at which time you’ll simply flip the carton upside down gently so as not to break any of the eggs. Do this every month to maintain the integrity of the egg yolk.
For storage up to three months, which is a bit longer than the trip in your question, regular room temperature is good.
For 6-9 months, a cool dark area is recommended, with temperatures about 65° - 68° and 75% humidity.
For extra long, up to 9-12 months, store in the refrigerator.
In case you're interested, I found the result of an experiment done by Jamie, of Prepared Housewives. She treated a carton of 18 eggs with mineral oil according to the same method described above. She put those in the refrigerator, making sure to flip them every month, for a year. She also put a carton of 18 untreated eggs right beside them. At the end of the year, she published her findings in Preserving Eggs with Mineral Oil, 1 Year Later.
As you said, the mark of a healthy egg is when it sinks to the bottom. She started there.
Mineral oiled eggs:
All 18 sunk straight to the bottom, except one, which had a crack in it.
All 18 floated on the top. One of those also had a crack, with brown sediment oozing out of it.
How well did they cook?
Mineral oiled eggs: Looked perfect and tasted great.
Untreated eggs: Cooked flat like a pancake. The texture was also different, slightly rubbery.
The linked page has pictures of eggs during each phase of the procedure. The final picture below shows the dramatic difference after cooking them over-easy. She ate the treated egg.
I can now say I've eaten eggs over a year old!