Does the DoD/DoN have any instruction or guidance that requires the use of
MIL-STD-961 or equivalent when writing a performance specification?
Yes, DoD issued DoD Manual 4120.24 in 2000 and two changes to this manual after the original version. The original version of the manual states "Defense specifications shall be prepared in accordance with MIL-STD-961" in Appendix 6. Refer to this manual for design specs.