No. I don't think they are better for eyes.
The designer brand sunglasses usually cost hundreds of dollars even more, the cost is not for the quality of the sunglasses, but for the famous brands. You know, most of the designer brands products are expensive. So, if you want to shield eyes and saving money, you needn't buy designer brand sunglasses. A pair of good quality sunglasses is enough.
In my view, I don't think the designer brand sunglasses are much better for the eyes. As we all know the designer brand eyeglasses are too expensive not only for his high quality but also for his famous brand. A well-known brand has to cost thousands of million dollars to advertise his brand, to find super stars to speak for him. These all costs will cost on customers. So if you spend the same money on a pair of glasses with different brands, the quality of the designer brand glasses may not as the same as the ordinarily brand's. In a word, if you want to save money and protect your eyes, you should find the best suitable glasses for you.
As a matter of fact, there are almost no real differences between designer brand sunglasses and normal sunglasses. Perhaps the only difference is that the former one give you a sense a superiority. Consequently, you have better confidence and smiles. On the technical level, maybe the former one also can be better in terms of quality, shape, as well as comfort. But a cheap but good quality one is able to meet your needs.