I have a hp color LaserJet CM1312 MFP series PCL 6 printer and when I print, it makes this squeaking noise.?

Buy some earplugs, if it ain't broke don't fix it :) Squeaks are usually an annoyance not a sign of something failing, they have a lot of plastic and nylon bushings in small printers that can squeak. If it starts grinding and you find pieces of chewed up plastic then start to worry. It may last for ever squeaking the whole time.

Unless it's only making paper jams, the problem might be from inside the printer. Now the other problems to consider are different stories. There are 3 things that can cause blank pages. The most common is the laser shutter pops off on the spring side and blocks the laser from hitting the cartridge. The shutter opens when the cartridge is inserted. The cartridge guide that sits above and behind the cartridge misteriously gets eaten by the printer. Without it, the cartridge rides too high for the toner to transfer. The high voltage power supply goes totally dead. If the first 2 are ok, then replace the high voltage power supply. Now for the paper jam+blank page issue. Sounds like you may have a selenoid flapper issue. Try printing from the manual feeder and see if it exhibits the same behavior. If it prints ok from the manual feeder, then the issue is probably the selenoid which controls the movement of the paper tray pickup unit.