  In this paper, we generalize the push-forward (Gysin) formulas for flag
bundles in the ordinary cohomology theory, which are due to Darondeau-Pragacz,
to the complex cobordism theory. Then we introduce the {\it universal quadratic
Schur functions}, which are a generalization of the (ordinary) quadratic Schur
functions introduced by Darondeau-Pragacz, and establish some Gysin formulas
for the universal quadratic Schur functions as an application of our Gysin
formulas.
