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?
Yes. Alternatively, you can deduce the computation of $H^*(BU;R)$ from the computation of $H^*(BU;\mathbb{Z})$, since the canonical ring-homomorphism $H^*(BU;\mathbb{Z})\otimes R\to H^*(BU;R)$ is an isomorphism (this is true more generally for any space whose homology with coefficients in $\mathbb{Z}$ is free and finitely generated in each degree, by the universal coefficient theorem).