The quick answer to your question is "no".
There is the theoretical, and there is the practical.
The literal, theoretical answer to your question lies in the MB requirements. Do the requirements for the Cooking MB anywhere in them state that the scout must be a First Class scout before taking the MB? No, they do not. And so the answer to your question is "no". For a counter-example, look at Lifesaving MB which requires a scout to first, "Complete Second Class rank requirements 5a through 5d and First Class rank requirements 6a, 6b, and 6e."
Also on the theoretical side, the SM is given some level of discretion in permitting any particular individual scout from taking certain MBs. For example, Lifesaving MB (again) requires as scout to, "Recover a 10-pound weight in 8 to 10 feet of water using a feetfirst surface dive. Repeat using a headfirst surface dive." If a scout is a non-swimmer, or a poor swimmer who is slightly built, the SM might decide that Lifesaving MB is not a good choice at this point in his scouting career.
Also in theory, requirements over and above those stated in the national program cannot be "added" by any individual or unit. That includes MBs and earning them. This circumscribes the SM's discretion described in the paragraph above. So, what he cannot do is say something like, "A scout must be First Class to work on Eagle-required MBs."
Finally, a unit may NOT artificially retard, inhibit or hamper a scout's advancement. Every scout advances at his own pace. This is closely tied to the rule that units and leaders cannot
So, now let's talk practice.
Every troop I have ever seen that says something like, "A scout must be First Class to take Eagle required MBs" is trying to intentionally retard the scouts' advancement. The goal is to make sure that they don't have any 14 yo Eagles, or some such other prohibited rationale. Every single time without exception. And if you look at it more, I am willing to bet that other aspects of their policy will have that same effect. They also typically control the Positions of Responsibility for the same reason and with the same effect.
As a practical matter, this is not necessarily all bad. I agree that most 14 yo Eagle scouts aren't really ready and that they typically don't get the most out of scouting that they can. They typically also drop out of scouting instead of giving back, running off to check yet more boxes and add yet more things to their "college resume". Also, Eagle-required MBs are indeed usually a bit more intensive and require more effort, which means that it is more likely to he will earn only a partial at Summer Camp rather than the whole badge. And, he can focus more closely on earning First Class if he's not working on MBs.
But it is not fair to the scout and it perverts the program. The justifications in the previous paragraph also prevent the scout from learning valuable life lessons about over-reaching one's abilities and overcoming failure. These things, too, are part of the scouting program.
Whether this is a problem for you and your scout depends on you two. Perhaps you are willing to put up with it because the program is so good. Or, perhaps it really meshes well with your scout's personality.
I will tell you that the few times I have seen scouts and their parents push on these points it has never ended well for anybody. The scout usually ends up transferring troops anyway.
In my opinion, good first year at camp MBs include Swimming, First Aid, Leatherworking, Wood Carving, Indian Lore, Horsemanship, Forestry, Geology...lots of good ones.