Brown eggs come from bigger hens than white eggs. Those bigger hens, usually Rhode Island Reds, require, because of their size, more food and therefore they are more expensive to maintain. The eggs however, white or brown, have the same nutritional value, cookablity, and flavor, but for whatever reason, some people still prefer brown eggs.
As for me, I prefer eggs that come from free-range, humanely cared for hens. Those eggs are the most expensive of all, brown or white, but the expense it well worth it in terms of a clear conscience and a relatively happy hen.
One reason that brown eggs are more expensive is because there is strong demand for them even at a higher price than that of white eggs. This is probably because brown eggs look "greener" even though there is not any difference.
The other reason why brown eggs are more expensive is because they are typically produced by hens that are more expensive to raise than hens that lay white eggs.
Because they are more expensive to produce, they cost more. Because people will still buy them at the higher price, they continue to be produced and sold.
The prices of any product is determined primarily by two factors - the cost of production and demand for product. Higher the cost of the product higher is the minimum price charged by the manufactures. No manufacturer will produce and sell a product if the product has to be sold at a price lower than its cost.
The demand of a product, the other factor determining price, is itself determined by the utility offered by the product. And when we talk of utility, we need to consider the utility to individual consumer as well as the number of consumer who derive the utility. For example, in India, brown eggs are considered more nutritious and also less allergic. Because of this brown eggs are considered more suitable for people suffering from some specific medical problems. Among general public, only people with lot of money are willing to pay the extra price for the brown eggs. People with limited financial resources prefer to by the more economical variety. Thus some people are willing to pay considerably higher price for brown eggs, but the total demand is quite low.
From the above discussion it is clear that brown eggs are priced higher becaause the cost of production of these eggs is higher, at the same time some people believe that they derive higher utility from them and, therefore are willing to pay higher price.
It is interesting to note that if the total demand for brown eggs was considerably higher, it may have induced producers to produce much larger quantities of brown eggs at lower unit cost, resulting from economy of scale. This in the end could have led to the prices coming down, rather than increase, with increasing demand. We can see phenomenon like this in many industries. For example in India, the prices of mobile phones has been coming down steadily with increasing demand.