Unfortunately Segre class doesn't share very good inclusion-exclusion property. For example, Let $Y$ ve irreducible smooth, and let $X=\cup_t C_t\subset Y$ is a union of curves. Assume that the intersection of curves are only nodal type point (no three points collapse together, and no tangent points). Then we have the following $$ s(X,Y)=\sum_t s(C_t,Y)+\sum [C_a\cap C_b] . $$ For general case, it will get only more complicated. For a detailed discussion, you can read this paper by Aluffi (https://arxiv.org/pdf/math/0203122.pdf)

Unfortunately Segre class doesn't share very good inclusion-exclusion property. For example, Let $Y$ ve irreducible smooth, and let $X=\cup_t C_t\subset Y$ is a union of curves. Assume that the intersection of curves are only nodal type point (no three points collapse together, and no tangent points). Then we have the following $$ s(X,Y)=\sum_t s(C_t,Y)+\sum [C_a\cap C_b] . $$ For general case, it will get only more complicated. For a detailed discussion, you can read this paper by Aluffi (https://www.math.fsu.edu/~aluffi/archive/paper165.pdf)