1-Have you tried disabling BOTH Squid and DG to see if the problem disappears? If you have tried this, does the problem resolve itself?
2-Have you tried enabling just squid to see if in fact it is just squid causing the problem. It just seems strange that you seem to be the first one with this problem as EA games are extremely popular.
Ok not entirely sure if I succeeded in this, squid does not seem that easy to disable? Or I have failed to locate the simple instructions.
I did this..
/sbin/e-smith/db configuration setprop squid Transparent no
/sbin/e-smith/db configuration setprop squid status disabled
/sbin/e-smith/expand-template /etc/squid/squid.conf
/etc/init.d/squid restart
service dansguardian stop
multiple attempts of the Registration process again, still resulted in the /squid/access.log..
1220823969.891 1 192.168.37.4 TCP_DENIED/400 1824 NONE error:unsupported-request-method - NONE/- text/html
1220823972.119 1 192.168.37.4 TCP_DENIED/400 1824 NONE error:unsupported-request-method - NONE/- text/html
1220823987.479 0 192.168.37.4 TCP_DENIED/400 1489 OPTIONS / - NONE/- text/html
1220823987.492 0 192.168.37.4 TCP_DENIED/400 1489 OPTIONS / - NONE/- text/html
1220823987.506 0 192.168.37.4 TCP_DENIED/400 1489 OPTIONS / - NONE/- text/html
1220823987.519 0 192.168.37.4 TCP_DENIED/400 1489 OPTIONS / - NONE/- text/html
FYI..
[root@l1nuxsvr ~]# /sbin/e-smith/db configuration show squid
squid=service
EnforceSafePorts=no
SafePorts=21,70,80,81,119,210,443,563,980,1024-65535
TCPPort=3128
TCPProxyPort=80:3128
Transparent=no
TransparentPort=3128
access=private
status=disabled
[root@l1nuxsvr ~]# service squid status
run: /service/squid: (pid 17112) 945s, normally down; run: log: (pid 3551) 209982s
[root@l1nuxsvr ~]# service dansguardian status
No DansGuardian process found.
pfloor
Look at the bright side... You can tell your son that SME is so smart that it knows that he hasn't done his homework and/or chores and to stop playing video games and go do his work. This will give you some time to fix it and when the problem is solved you can tell him "Looks like you have finished your chores, lets see how smart SME really is". He logs on and Bingo, the game works and you look like a genius 
I love the theory

He already knows SME is smart enough to alert me to inappropriate web surfing, and self learning that attempting the next day finds the site is blocked. So SME keeping track of his chores and homework is not a huge extension on this! (that said he is 13).
Reminds me when he was very young, I had a basic LAN between 2 PC's and the cable running across the floor, he was playing Lego Loco and sent the train into the tunnel so it would appear on my PC.. He crouched over the cable on the floor, and said "I can hear the train in the tunnel dad !"
Innocence is so cute !!