I have three deBuyer Mineral B pans in different sizes. They're great.
You and Kokopuffs are both right. The problem is with the seasoning.
Carbon steel and cast iron pans need repeated seasoning and very gentle seasoning before developing a truly non-stick surface (but the seasoning oil doesn't always have to be bacon). There are plenty of good ways to start a season. The potato method is one, but not the only way. Starting a season isn't a big deal, but killing it is. You must protect the nascent seasoning by not scouring it off when you clean the pan. It takes care and patience to develop the season to the point where the pan becomes non-stick. Beyond patience, you should also recognize that even once a carbon steel or cast iron pan is well seasoned, it will not function in the same way as a teflon, ceramic or other artificial non-stick surface.
For one thing, a season requires appropriate maintenance, which includes oiling before putting away. For another, cooking requires a little more fat.
It's hard to know what went wrong with your eggs. From your description, I think there were more problems with suiting your technique to the pan than with the pan itself.
Let's talk about cooking those scrambled/omelette breakfast eggs in a way which will guarantee success. Preheat the pan thoroughly over medium-low heat. DeBuyer Mineral pans are fairly thick and require some time to pre-heat. Don't rush it. Add a little bit of vegetable oil to the pan, and allow it to heat to the shimmer point. If you like some oil in the butter for flavor or to help prevent the butter from browning, add the butter to the oil. If not, dump the excess oil into the sink. After a few months, you probably won't need to oil the pan to ensure a slick surface before melting the butter; but it's good technique anyway.
The pan should be hot enough so that the butter foams when it hits the pan, but not so hot that the butter immediately browns. Allow the foam from the butter to begin subsiding before adding the eggs to the pan.
Give the eggs a few seconds to begin to set. With the pan on the fire, take the handle and swirl the pan; neither too diffidently nor with too much enthusiasm. If the "system" is working, a clump of curds will form in the center of the pan and swirl around in the un-cooked liquid eggs as you continue to swirl the pan. It will form the basis of your omelette or scrambled eggs. After you have the clump going, you can use a spatula if you like for the processes of lifting the edges of the solids and getting liquid eggs to the pan surface; but as the pan's season and your egg-cooking skills develop, you'll find that you really don't need it.
After cooking, clean your pan. Just as a matter of improving the season, the best way to clean is by wiping the pan with a dry paper towel or a damp sponge. Often though, you'll need to at least rinse with plain water. Mild dish soap isn't a disaster, but it makes the process take longer. If there's stuff stuck to the bottom, which requires you to use water, soak the pan first to soften it. Use a soft dish brush.
Dry you're going to have to dry the pan over moderate heat; after the pan is dry, but still on the fire, add just enough oil in the pan to swirl it around; heat the oil to the shimmer point; pour the excess oil out of the pan; wipe the pan with a paper towel, making sure to oil the entire interior with the remaining oil; and finally allow the pan to cool completely before putting it away.
Oiling the pan with very hot oil every time after it's washed will help improve the season. Oiling the pan as part of the pre-heating process will help kick-start whatever non-stick properties it's already developed. Using techniques which keep food from sticking to the pan, will make clean up easier and less problematic for the season.
This is a picture of a Mineral pan which is about half way on the way to a good season (ignore the pepper mill):
The surface of the pan is shiny because it's oiled. ALWAYS oil your carbon steel and cast iron pans before putting them away.
This is a picture of my granddaughter, Ashlee.
Her surface is shiny because she's so new. She has nothing to do with any of this, but I couldn't resist.
Of course eggs aren't the only things you'll be cooking in the pan. If you sear meat in your pan, it will stick during the first part of the searing process even if the pan is oiled. That sticking is a part of forming a good sear (aka crust). Shake the pan every minute and after the sear has formed, the meat will naturally "release" and move easily -- without prying it from the pan with a spatula. Allowing the meat to sear until it releases will create a fond which can and should be lifted off the bottom of the pan by deglazing. Levering meat off the pan, will leave sticky stuff which will only clean off the pan with difficulty.
Hope this helps,