There are $n$ values $y_1,y_2,...,y_n$ where $y_i\in \mathbb{R}$ and $Y=\min_i\lvert y_i\rvert$. Also $Z=\Sigma_{j} \left(2x_j-1 \right)Y$, $x_j\in \{0,1\}$. Is it possible to establish a relation between $Z$ and $Y$ like range of $Z$ in terms of $Y$ or any relation(s)?
There is no specified relation among the $x_j$ and $y_j$. In particular, in general $(2x_j-1) \neq \text{sign}(y_j)$.
As written, the question makes little sense because $Y$ can be factored out and the question boils down to giving bounds on $\sum_i\pm1$ (obviously $\le n$ in absolute value, which is tight.)