I can think of a few ways to do this with a smart contract, but in bitcoin it's a bit harder.
I'm not sure about the implementation details, but I think the general logic would be as follows:
1) They sign a timelocked transaction using their private key, sending the coins to the target address but not redeemable until date x.
2) The timelocked transaction is stored on your server
3) Before date x arrives, they move their coins to a new address and sign another timelocked transaction using the private key of the new address.
4) Rinse and repeat until date x arrives when your server publishes the timelocked transaction to the network.
This way their private keys never touch the server, they can spent their coins however they like and the owner of the receiving address can't spend the coins until the dead man's switch has triggered.
This is basically the solution but it's kind of redundant.
A simpler one would be to just sign the tx that would spend all his coins right now. And store that transaction on a server. Write code in your favourite language that broadcasts the tx after Y amount of time just for an added layer of security. And open up a port on your server where the application can listen to.
If the application doesn't get pinged once every X months, weeks, whatever, then it calls the function, and after Y amount of time, the tx will be broadcasted.
So he has to ping the server every X interval, and if he somehow fucks up and forgets, he has Y more time to stop the application from broadcasting his coins.
Hell if you want I can probably set this up for you in node.js right now.