There 4 different sizes of flat-rate boxes. There are small (1), medium (2), and large. Don't even bother with the 'small' or 'medium' flat rate boxes, they are basically worthless, I don't know what anyone would use them for!
**Clarification: there are 2 medium boxes, one of them has a dimension of 3 or 4 inches on one side. Not good for shipping eggs, not enough room.**
I use the #4 or the #7 boxes, both available from the usps site for free. The #4 box is 7x7x6, and is great for a small amount of eggs. the #7 is 12x12x8, so it holds a lot. I used the #7 just the other day to ship 100 quail eggs to Alabama from Kentucky, it cost about $7 for the shipping. I can fit probably 4 dozen LF chicken eggs in the #7 box.
The #4 is good for 6-10 LF chicken eggs, 8-12 bantam eggs, or 20-30 quail eggs. The shipping is based on how much they weigh, and where they are going. Anything weighing less than 1 pound, going anywhere in the continental US is about $5 for shipping. That's the same price as the small flatrate box. I sent a package today in the #4 box that is going all the way to Oregon. The package weighed 2 pounds and was also about $5 for the shipping.
I hardly ever use the flatrate boxes. The only times they would save my customers on shipping is when the boxes themselves are too small to hold everything. The #7 box is much more roomy, even more than the large flat rate box. I believe the size of the large flat rate one is 12x12x5 or so, where the #7 is 12x12x8.
Just my 2 cents.