My understanding of "the squeaky wheel gets the grease" is "If you don't complain, your situation will not change for the better".
In this sense, you can say 「不平則鳴」(express your opinion when unfairness occur) is similar to it.
Another answer found here is「愛哭的孩子有糖吃」 seems like an direct translation of a similar English expression "It is cry babies who get candies". Not much different from 「嘎吱響的輪子先上油」(The squeaky wheel gets the grease) in sentiment.
Generally, traditional Chinese people are more reserved, less likely to complain loudly compare to Western thinking people. That's why ideas like "The squeaky wheel gets the grease" or "It is cry babies who get candies" were kind of foreign to us.
We agree with the expression 「先打出頭鳥」(the bird that sticks its head out gets shot first) more. Japanese also have a similar expression "出る釘は打たれる" (the nail that sticks out gets hammered).