Is there a requirement for self-locking / positive locking electrical connectors for aircraft systems?
EDIT: Note that the original question, which this answer primarily addresses, did not specify electrical connectors.
The FAA does provide regulations which at least partially cover this subject, though the guidance is rather broad.
The airworthiness standards in 14 CFR 23.607 for normal, utility, acrobatic, and commuter category airplanes stipulate that:

(a) Each removable fastener must incorporate two retaining devices if the loss of such fastener would preclude continued safe flight and landing.
(b) Fasteners and their locking devices must not be adversely affected by the environmental conditions associated with the particular installation.
(c) No self-locking nut may be used on any bolt subject to rotation in operation unless a non-friction locking device is used in addition to the self-locking device.

Likewise, the  airworthiness standards in 14 CFR 25.607 for transport category airplanes stipulate that:

(a) Each removable bolt, screw, nut, pin, or other removable fastener must incorporate two separate locking devices if—
(1) Its loss could preclude continued flight and landing within the design limitations of the airplane using normal pilot skill and strength; or
(2) Its loss could result in reduction in pitch, yaw, or roll control capability or response below that required by Subpart B of this chapter.
(b) The fasteners specified in paragraph (a) of this section and their locking devices may not be adversely affected by the environmental conditions associated with the particular installation.
(c) No self-locking nut may be used on any bolt subject to rotation in operation unless a nonfriction locking device is used in addition to the self-locking device.

As you can see, these regulations do prohibit certain types of locking fasteners in certain circumstances. For example, a nylon self-locking nut is not acceptable to fasten the exhaust manifold to the cylinder head; an all-metal self-locking nut would be acceptable, however.
The greater concern for a system such as a headset may be that to install it with fastening hardware—screws for example—would be to make it an installed part of the aircraft. The installation would then be subject to the various regulations of an installation: a log book entry, weight and balance considerations, etc.
Additionally, AC 43-13-1B, Ch 7, provides non-regulatory guidance regarding use of fasteners, including the use of locking and safety mechanisms. I will note that this guidance is superseded by manufacturer maintenance data for most modern aircraft.