Why is the fridge water line popping out of the connection when under pressure?
Make sure that the pipe is inserted into the valve as far as it will go before you tighten the compression fitting - it would only slip out if not properly tightened against the ferrule.

Two other recommendations:


Get rid of the piercing saddle valve and install a ball valve (1/4 turn) that has either a compression fitting or is soldered in place.  
Scrap the plastic pipe and pick up some 1/4" flexible copper supply line.


The first is more important.  The piercing values are usually not to code.  They can be difficult to turn off (not good in the event of a leak), and often the pierce does not make a  large enough of a hole for adequate water flow, which can result in ice making problems and slow water dispensing.