Your meter (I assume it is a multimeter) is likely to output about 2V in the diode check mode. That is enough to faintly light up a white or blue LED, but as the LED needs about 3 volts the meter cannot measure it. The multimeter will be able to measure a red, yellow, orange or green LED as these only require about 1.8V.
LEDs have a polarity. As explained by 807 and based on your description, the 2 LEDs are likely wired in inverse-parallel. Also this is something you have to consider when testing the LED with the multimeter. The LED should light up in one way but will not light up when the test leads of the meter are reversed.
You can replace a broken white LED by another white LED or a blue one. About any 5mm white or blue LED will do. It is likely that the originals are high intensity LED's. Due to the voltage difference it is generally not possible to replace a white LED by a yellow, orange or red LED unless the resistor is adapted to reduce the voltage over the LED.