We report an observation of the B-+/- -> J/psi eta K-+/- and B-0 -> J/psi eta K-S(0) decays using 772 x 10(6)B (B) over bar pairs collected at the gamma (4S) resonance with the Belle detector at the KEKB asymmetric-energy e(+)e(-) collider. We obtain the branching fractions B(B-+/- -> J/psi eta K-+/-) = (1.27 +/- 0.11 (stat.) +/- 0.11 ( syst.)) x 10(- 4) and B(B-0 -> J/psi eta K-S(0)) = ( 5.22 +/- 0.78 (stat.) +/- 0.49 (syst.)) x 10(-5). We search for a new narrow charmonium(- like) state X in the J/psi eta mass spectrum and find no significant excess. We set upper limits on the product of branching fractions, B(B-+/- -> XK +/-) B(X -> J/psi eta), at 3872 MeV c(-2) where a C-odd partner of X(3872) may exist, psi(4040) and psi(4160) assuming their known mass and width, and over a range from 3.8 to 4.8 GeV c(-2). The obtained upper limits at 90% confidence level for XC-odd(3872), psi(4040), and psi(4160) are 3.8 x 10(-6), 15.5 x 10(-6), and 7.4 x 10(-6), respectively.