OK, well the squirrel traps are really cheap and easy (just a simple cage trap - loads for sale on Ebay). The method my keeper friend uses for dispatching is to place a hessian sack over the trap door, then encourage the squirrel into the sack with a wee prod, then choose your moment and club it very decisively over the head. OK, not pretty, but humane enough IMHO.
The magpies are a bit more tricky. The trap used is called a "Larsen trap", and is usually 'baited' using a live magpie in a separate compartment (please don't ask where you're supposed to get one of them from - it's kindof an impossible question!
). The idea is that the resident magpies think the caged stranger is encroaching on their territory, and then try and drive it away, and in the process, get caught.
That said, if your magpies are in the habit of taking hens eggs, you can 'bait' the trap with a fake hens nest and eggs (break one egg so that the yolk is showing), and that would probably work, without the need for a caged 'call bird'. (Yet another way is to use a dummy plastic magpie, so if you have no luck with the eggs, PM me, as I might just have a spare one lying around somewhere
).
The most important bit is that you MUST obtain a registration number from your local police wildlife officer before you can use one of these traps. This is free to obtain, and then you must then put the number on the trap, along with a message that says something like "this is a legal trap. If you have any concerns about the way it is being used, please contact PC Plod, #1234 on 01234 567 8910", quoting trap reference 5555." You MUST then check the trap regularly, and ensure any caged bird always has room to spread its wings, protection from the weather, and a supply of food and water.
The best sources of information on these traps are places like the BASC website, and somewhat ironically, the "
Action against Corvoid traps" website. Naturally, these guys are totally against the use of Larsen traps, but their website makes good reading, and it certainly helps to be informed about both sides of the debate.