I am reading this post, Prop. 2.1. It seem that none of the argument is dependent on the we are working with coefficient in $\Bbb Z$.
Hence, let $R$ be a unital commutative ring.
(I) Do the results for $H^*(BU, \Bbb Z)$ hold for $H^*(BU;R)$?
(II) Are the arguments the same?