What was Friedan's problem in The Feminine Mystique? Doesn't it only apply to a limited group of bored, white, middle class women?

The problem that Betty Friedan describes in her 1963 work, The Feminine Mystique, is the dissatisfaction of suburban housewives. She found that many of her former classmates had gotten married, had children, and lived in comfortable circumstances in suburban settings, but were still unhappy and unfulfilled. These women felt restricted in their life prospects and as though their personal interests and values were second to their duties as wife and mother. Indeed, Friedan found that many women abandoned personal interests in order to pursue the "feminine mystique" of being fulfilled by a role as wife and mother. Friedan is credited with sparking the second-wave of feminism in the United States with this book, though I would argue that many women in the present period feel that they ought to chase the happiness promised by marriage and motherhood, even if it means sacrificing their personal interests and values. 

You are right to assume that this problem was significant for white, middle class women and somewhat irrelevant to women of other classes and racial backgrounds. In fact, this has been one of the biggest criticisms of Friedan's work and the second-wave of feminism in the United States-- it leaves a lot of women out of the discussion. Bear in mind that Friedan, herself, was a white, middle class woman, so her social circle and academic concerns naturally mirrored this. At the time Friedan was writing, lower class women and women of color were not offered the luxury of being stay-at-home wives and mothers. Many women, especially women of color, were employed in domestic service or had no option but to be a wife and mother. There's a big difference between a middle class, educated, white woman choosing to forsake her personal interests in order to be a wife and mother and a woman of color being denied employment outside of the home on the basis of her skin color, class, or level of education.

Upper class woman (who were essentially always white) were also offered the luxury of choosing not to work outside the home, and were able to hire childcare and domestic servants. An upper class woman in Friedan's time might have the title of a stay-at-home wife and mother but need not have done any of the work. Such women had the time and money to pursue their personal interests if they chose to, in contrast to the middle class woman who had to choose between motherhood and personal development.

In short, yes. Friedan's book is entirely focused on the narrow portion of society made up by white, educated, middle class women who had become disenchanted with the lifestyle they believed would offer them happiness. Criticisms of Friedan's highly exclusive work, and other second-wave works, have since sparked the third-wave feminist ideology of inclusiveness of women of color, trans women, and women of all classes.