No, I don’t think they will say no to a shadchan about a girl who will not support him completely (for the first few years, depending on how long the boy plans to learn), but as with many Yeshivos, it is preferred. Very often, CC boys will get Smicha and become a Rosh Yeshiva pretty soon after (sometimes out of town).
WolfishMusings, I don’t think the Rosh HaYeshiva was dictating financial terms for dating a boy, just that most of the boys are looking for that. CC encourages support from a wife rather than parents.