Think one would have to disclose firewall vendor for anyone to begin to make suggestions there. More than likely, if you are sure it's not a Moodle caching issue, someone will have to contact the firewall support/provider.
But, It might not be just the school firewall ... that's the end point from your connection to Moodle outside of school. Example: have a home connection that uses OpenDNS added to their 'firewall' for home customers. Sometimes (seldom, I might add) OpenDNS has determined (somehow) that a school site is suspect and I cannot hit the school site.
There could also be issues with network ... network has to work before application. Suggest doing a traceroute from your home location to the school moodle. That may not complete all the way to the moodle server, but you might discover there is an issue with routing (which you cannot control). If that is the case, carriers have to get involved.
And there is a situation like this ... had a Moodle Linux server running iptables (server firewall). Similar issues with teachers/students attempting to access the Moodle from the outside turned out to be in-compatibility with the schools firewall and iptables. If we turned off iptables on the server, no problems. We did that until the school replaced their firewall with a better product ... which was also compat with Moodle servers iptables.
'spirit of sharing', Ken