Well, they're not completely wrong, I'll give them that.
A Warning on Wild Rabbit Welfare
If you're dead set on having a cottontail as a pet, you have to be a special kind of crazy, in which case no words would ever deter you...but I'd recommend looking into similar legal alternatives such as the San Juan rabbit, bred for training hunting dogs.
A Bunny is a Bunny, Right?
Let's start with the basics. The eastern cottontail (Sylvilagus floridanus) and domestic rabbit (Oryctolagus cuniculus) have very fundamental differences, which you can determine just by the fact that they are not even a part of the same genus. In fact, the domestic rabbit is still considered to be of the same species as the European wild rabbit that it derives from. While both share the family Leporidae that defines them as rabbits, they are not quite analogous.
If you were to find a confused-looking agouti rabbit hopping about in your backyard, the easiest way to decipher whether or not it's meant to be there is by looking at the head. Compare the top two images above; the cottontail (right) has a narrower, more triangular face and large round eyes set wide on the face.
However, my personal go-to for figuring out if a rabbit is domestic is by looking at the ears. The European wild rabbit, as well as its friendly descendant, has much longer ears that are much more slender along the entire length, and thicker (especially at the edges). This is what the stereotypical rabbit ear looks like. On the other hand, the ears of a cottontail are so thin that light easily passes through them, as seen above, and are much shorter and rounder. When Squirrel started growing into her adult form, I was actually shocked at how small her ears were, and wondered if there was something wrong with her...until I paid more attention to her relatives in my woods.
But that's not all...
In addition, the pictures on the right show the almost terrifying contrast in leg structure between the two species. (The stretching cottontail is NOT photoshopped. I've seen Squirrel do that after getting up from a nap while arching her back and yawning...nightmare fuel.) In many ways, the cottontail is like a well-bred racehorse when compared to the sluggish, lumbering domestic. Those supermodel legs allow the wild rabbit to run at 18-25mph for up to half a mile, zigzagging the entire way to avoid predators...an incredible feat for such an often-overlooked creature!