This work investigates the eigenvector distribution of Gaussian random matrices with finite-rank deformations in the critical regime of the Baik-Ben Arous-Péché (BBP) transition. For both rank-one and higher-rank deformations in the GOE and GUE, as well as the rank-one general β-ensemble, we show that the squared overlap between the leading eigenvector and the spike direction, rescaled by N^{1/3}, converges to the negative reciprocal of the derivative of the associated Airy-Green function. The analysis relies on eigenvector-eigenvalue identities, soft-edge asymptotics, and properties of Airy-Green point processes, providing a unified representation of the critical BBP eigenvector distribution and its natural extension to general β-ensembles.