During flight over mountainous terrain, the pilot encountered "horizontal snow squalls." He reported that he flew through the snow squall for about 25 miles while maintaining ground contact. He approached a cloud layer, but was able to fly over it at 16,500 ft MSL. As he continued, the engine began running rough. With the temperature at -15 deg Fahrenheit (below yellow arc on carb air temp gauge), he was reluctant to use the carb heat. After switching tanks and mags with no improvement, he applied full carb heat. However, the engine continued to run rough and finally stopped. He entered the clouds, and with no vacuum suction, descended using needle, ball and airspeed. He turned into the wind to reduce the ground speed, and at about 10,000 ft, the aircraft crashed and came to rest in trees. The ELT operated, but the antenna was broken. The pilot made a shelter, remained overnight and was rescued on the following afternoon. No engine problem was found except the plastic foam induction air filter contained water when checked.