NO Think again, the high order bit of a DDS output is very definitely
not a squared up version of the fundamental !  Used on its own it would
have a spectrum not totally unlike noise.

I thought again and, yes, the high order bit of the phase accumulator is not
the squared up version of the sine output, but, after the phase accumulator,
a DDS has a sine lookup ROM. What I really meant was the high order
bit of the output of this ROM. Using this bit, I feel confident to state again
that you don't need a DAC to generate a square wave of the wanted frequency.

With the AD9850 you have to use the comparator, as you do not have external
access to the output of the sine ROM.
You are right saying that a PIC probably doesn't have enough horse power
for this task. Maybe some chips  of the Scenic family ???

And, while the AD983x may be inexpensive, they are nevertheless SMD devices,
not particularly easy to solder (what an understatement...!)

73  Alberto   I2PHD