We use the connective formal group law to define a one-parameter ($β$-)deformation of the motivic Segre classes of Schubert cells in the $d$-step flag variety. This $β$-deformation specializes to the motivic Segre classes of Schubert cells when $β=1$ and to the Segre-Schwartz-MacPherson classes of Schubert cells when $β=0$. We define rational function representatives for the $β$-deformed classes in the $d=1$ case in terms of a solvable lattice model, and we prove a combinatorial formula for the structure constants in the $β$-deformed basis in the $d=1$ case using Knutson-Tao puzzles. The proof of the puzzle formula involves intertwiners for representations of the multi-parameter quantum group of type $\widehat{a}_2$. We show that our $β$-deformations can be viewed as quotients of canonical elements in a quotient of the equivariant algebraic cobordism ring of the cotangent bundle of the flag variety by proving that the canonical elements satisfy a GKM type condition.