Name for the need to reply “you can’t fire me, I quit”
I don't know of any term that refers specifically to this sort of counter-offensive reaction to ending a social relationship (be it romantic, professional, or otherwise), but I vaguely recall reading about some theory that distinguished reactions to relational stress as follows:
$$\begin{array}{c|c|c|}&amp;\mathbf{Active}&amp;\mathbf{Passive}\\\hline\mathbf{Constructive}&amp;\rm Actively\ constructive&amp;\rm Passively\ constructive\\\hline\mathbf{Destructive}&amp;\rm Actively\ destructive&amp;\rm Passively\ destructive\\\hline\end{array}$$
The reaction in question is actively destructive. A passive alternative would be less decisive / assertive, and either patient / understanding / cooperative (i.e., constructive) or apathetic / pessimistic / negative (i.e., destructive). Sorry I don't have a reference off the top of my head; I think this came from my social psych textbook as an undergrad, but I'm not sure.

As for plausible explanatory constructs, consider:


Dominant personality
Avoidant or anxious-ambivalent (i.e., insecure) attachment style
Needs for power and autonomy
Behavioral tendencies involving defensiveness and reactance

Certain defense mechanisms may apply depending on whether the person is reacting genuinely or just defensively (and doesn't actually want to end the relationship).