Does v4.0.0 require full internet access

It seems v4.0.0 requires full internet access, as there are several widgets that can query external sites. Is it possible to run v4.0.0 behind a strict firewall with a squid proxy?

I am not sure about the squid proxy specifically, but access to is required for CryoSPARC function.
Did you experience that failure to access other sites, like the EMPIAR repository, impeded data processing functions?

Can you advise which hosts, ports, inbound or outbound are necessary to be open? I am running Cryosparc in a secured space and any outgoing traffic must be explicitly allowed.

Outgoing access to is required. Some optional CryoSPARC functionality based on internet resources can be disabled with the CRYOSPARC_DISABLE_EXTERNAL_REQUESTS variable inside cryosparc_master/ (details).