We report results from a search for the lepton flavor violating decays B0s→e+μ− and B0→e+μ−, and the flavor-changing neutral-current decays B0s→e+e− and B0→e+e−. The analysis uses data corresponding to 2  fb−1 of integrated luminosity of pp collisions at s√=1.96  TeV collected with the upgraded Collider Detector (CDF II) at the Fermilab Tevatron. The observed number of B0 and B0s candidates is consistent with background expectations. The resulting Bayesian upper limits on the branching ratios at 90% credibility level are B(B0s→e+μ−)<2.0×10−7, B(B0→e+μ−)<6.4×10−8, B(B0s→e+e−)<2.8×10−7, and B(B0→e+e−)<8.3×10−8. From the limits on B(B0(s)→e+μ−), the following lower bounds on the Pati-Salam leptoquark masses are also derived: MLQ(B0s→e+μ−)>47.8  TeV/c2, and MLQ(B0→e+μ−)>59.3  TeV/c2, at 90% credibility level.