Bases on the information you provided, it appears to me that the support tech has provided you with invalid reasons for not being able to set up the webcam.
It is not clear whether you are accessing WiFi via WEP or WPA. WEP is indeed an outdated protocol with serious security flaws. However, a non secure network is generally irrelevant to webcam setup.
Presence or absence of unauthorized users is generally irrelevant to webcam setup.
"Network Access Protection" is not a reason, but a technology for controlling network access based on system health of the host. In particular, "Administrators can configure health policies that make it possible to ensure that computers not in compliance with system health requirements have restricted access to the network." (Ref.: https://en.wikipedia.org/wiki/Network_Access_Protection) This is generally irrelevant to webcam setup.
"Agent-stopped net login stopped" sounds like a possible software configuration problem, but the expression contains a logic/grammar error, and as such, does not make sense. Google search for this expression has no non-trivial results. In my opinion, the validity of the expression "Agent-stopped net login stopped" is dubious, and relevancy of the expression is dubious.
I doubt that the support technician you spoke to is even qualified or has technical capabilities to perform diagnostics, which would have results similar to this. I would consider revising my opinion, if the technician could provide a better explanation, but the chance of the latter happening is next to impossible.
To summarize, what you have described sounds like an incompetent ruse with the purpose of having you pay for the additional services. Customer support and tech support occasionally cut corners in an unethical way, especially with outsourcing thereof to overseas personnel. Obtaining those services may or may not be helpful with the problem; the advantage to that would be that perhaps you would be serviced by more competent staff in case of Geek Squad, which might determine and help you fix the actual issue.
However, webcam configuration is generally a trivial procedure, and if for some reason it is blocked on the network due to software or hardware firewall settings, your best bet would be to have your local system administrator or tech support look at this. I refer to people who set up your network in the first place and/or usually perform its maintenance.
It is not very likely that the resolution of this issue would be possible or effective with remote support. Furthermore, resolution of the issue in this case seems to be probably outside of the competence scope of Logitech support.