  Given any order ideal $U$ consisting of color-squarefree monomials involving
variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional
simplicial complex $\Delta_{\mathrm{bal}}(U)$ that we call a balanced squeezed
complex. In fact, these complexes have properties similar to squeezed balls as
introduced by Kalai and the more general squeezed complexes, introduced by the
authors. We show that any balanced squeezed complex is vertex-decomposable and
that its flag $h$-vector can be read off from the underlying order ideal.
Moreover, we describe explicitly its Stanley-Reisner ideal
$I_{\Delta_{\mathrm{bal}}(U)}$. If $U$ is also shifted, we determine the
multigraded generic initial ideal of $I_{\Delta_{\mathrm{bal}}(U)}$ and
establish that the balanced squeezed complex $\Delta_{\mathrm{bal}}(U)$ has the
same graded Betti numbers as the complex obtained from color-shifting it. We
also introduce a class of color-squarefree monomial ideals that may be viewed
as a generalization of the classical squarefree stable monomial ideals and show
that their graded Betti numbers can be read off from their minimal generators.
Moreover, we develop some tools for computing graded Betti numbers.
