What are we seeing in the pics? If that is the subfloor, and the
subfloor is on the floor joists, then you will need at least another
1/2" plywood plus 1/4" backerboard. You need at least a total of 1
1/4" solid flooring for the tiles not to flex or break.
That's what I did in my kitchen. I ripped up the floor down to the
3/4" subfloor, put 1/2" plywood, then 1/4" backer on top. Don't forget
you have to thinset the backerboard as well as screw it to the
plywood. I'm not sure why he would want to use weather paper. If that
is an existing oak floor in the pics, then he probably wants to use
the weather paper to prevent the oak from getting wet, which would
cause the oak to expand, thus cracking the tile