the joy of parenting in a world of 24x7 on internet. they could quickly get around this proxy by retrieving the content via another proxy. there are plenty of those on the internet, quite happy to forward entire websites. they could also get another web browser or just change/remove current proxy setting. there's no way to specify a proxy system wide, it's set at application level. also whilst i've setup several squids myself without too much hassle, depending on your confidence and time available to read the details, you may want to avoid that.
what i think you actually want is some good parental controls software that is not visible to the user. there's plenty of that around, and they do have time based url restrictions and various other flexibilities. they hook into the tcp/ip stack, so all traffic is checked against specified rules. a special key sequence and assigned admin password brings up the gui to manage it. they also optionally log all traffic, so you can tighten the rules if needed.
but even these can be thwarted by an enterprising youngster, once they find out they're being watched & ruled 24x7 on internet by parents. hence the only real way is to properly monitor usage as a parent (physical visits to computer room) and early education about time management and the various dangers on the net. being careful not to be too negative/confrontational is key to such dialogue with kids.
the hardest line to type correctly is: stty erase ^H