The unitary Cayley graph $X_n$ has the vertex set $Z_n=\{0,1,2,\dots,n-1\}$ and vertices $a$ and $b$ are adjacent, if and only if $\text{gcd}(a-b,n)=1$. In this paper, we present some properties of the clique, independence and distance polynomials of the unitary Cayley graphs and generalize some of the results from [W. Klotz, T. Sander, \emph{Some properties of unitary Cayley graphs}, Electr. J. Comb. 14 (2007), \#R45]. In addition, using some properties of Laplacian polynomial we determine the number of minimal spanning trees of any unitary Cayley graph.