My 2006 ford Taurus will not stay running after my a/c quit blowing cold and making a squealing noise for about a week. How can I diagnose it?

probably low or no gas
until you have the system checked for leaks it will be a waste of money just adding gas
Take it to an accredited ac specialist and have it properly diagnosed

try this.
Diagnosing Belt Noise squeak or squeal on Serpentine Belt

The belt has most likely moved either in or out on the bottom pulley. Check it first. I had the same problem it looked ok but was out one groove missleading designed pulley.

Your water pump has broken down the seal has worn and let the water out seizing up pully the squeling noise is drive belt sliping as engin trys to turn pump replacement soon as,..... if u use it u will blow engine up

How old is your battery? If it is the original 2006 battery now is the time to replace it. A battery needs to be replaced after about 4 winters. Cold weather reduces the battery capacity and makes it harder to start. If you wait the car may not start at all.

Most likely cause is a bad automatic belt tensioner. See this post on how to check a belt tensioner

http://ricksfreeautorepairadvice.com/diagnose-a-belt-tensioner/

Then see this post on how to diagnose belt noise

http://ricksfreeautorepairadvice.com/diagnose-belt-noise/

Check your oil stick and make sure the oil is at the right level.

It may quiet down for a while but the damage is done so the pump will eventually need to be replaced.

sounds like heater control valve situated under scullle panel common fault on ka set heater fan switch half way between hot and cold with heater blowing then touch valve if pulsing or clicking noise it is faulty if doesnt solve problem faulty switch

change you pcv vavle and distrubter rotor