Simple. Just denote the squares 1, 2, 3, ..., 5n, and split into sets by their remainder when divided by 5. All partitions of this set are independent of the other sets, and all sets have the same amount of partitions, therefore if one of the sets can be partitioned in $N$ ways, the rectangle can be partitioned in $N^5$ ways.

Let $F_n$ denotes the number of partitions of rectangle $1\times n$ into unit squares and dominoes. Color cells of rectangle $1\times 5n$ by repeating blocks of $1,2,3,4,5$ from left. For every $i=1,2,\dots ,5$ move cells colored $i$ into one $1\times n$ rectangle $r_i,$ without changing of their order. Clearly parts of every broken domino have the same color, so we see that the number of partitions of $1\times 5n$ rectangle by squares and broken dominoes is the same, as the total number of partitions of all $r_1,r_2,\dots ,r_5$ by squares and dominoes, which is $F_n ^5.$