Here is a partial answer.  It proves a fault-free rectangle can be assembled from rectangles of size mxn where m and n are relative primes and m,n > 1.

For the case m,n not relative primes, you can divide m and n by the common divisor, but you  might end up with a dimension 1.

This means it actually proves the claim for m,n > 1 and one is not a multiple of the other.  The remaining cases are covered by the 1xn case solved earlier by Bubbler.

>! It is really simple.  Here is the solution for size 3x4.  
>!  
>! A central rectangle is surrounded by four large squares made of n times m rectangles.
>! The gaps at the corners are just the right size for stripes of rectangles.
>! It works for any size mxn.  
>!  
>! If m,n are relative primes and one is not a multiple of the other (i.e none is 1) then the lines are guaranteed not to align across black lines, making the rectangle fault-free.  
>!  
>! [![enter image description here][1]][1]  


  [1]: https://i.sstatic.net/bWw1G.png