Is there some black magic required to get higher order harmonics out
of an oscillator?

No black magic, you just need to filter for the fifth harmonic. I'm
extracting the 16th harmonic of 64 MHz out of a MMIC, and retuning the
filter, can actually extract a little more 15th than 16th. You can generate
a comb generator to do the job of generating large quantities of all
harmonics, but for this simple of a job, it'd be overkill big time. Either
make a better filter, amplify your fundamental more before filtering, (the 5
volt digital squarer ought to put out +17 dBm) or check the HP app note
AN983 and take advantage of "filter gain".

W4ZCB