I think your fan's ball bearings are not working well and you need to change them. If the fan stops working that may be because of failing capacitor but the squeaking problem indicates that the bearings of the fan are not working well and need to be replaced. Moreover grease is used in ceiling fans not oil.
Often the mounting is what is squeaking on many fans. Try taking the fan down and check all the mounting hardware and if you find the culprit use candle wax as a lubricant. Before taking the fan down though there are often two to three screws on either side of the fan mounting ring try snugging them up and see if that stops the squeak. Remember to turn the fan off first though and if you do need to take it down remember to turn off the breaker or remove the fuse so you don't get electrocuted.