Otdfbt

I've always countered with "the quacking duck gets shot".

Insofar as "the squeaky wheel gets the grease" is saying that people who complain get attention, I think this expression means roughly the opposite:

Good things come to those who wait

A fun/playful answer to "The squeaky wheel gets the grease" might be "It's better to be silent and be thought a fool than to open your mouth and prove it"

There are many versions of the "remain silent" saying. Some of them are documented by Quote Investigator.

I've always liked: "Empty Wagons make the most noise."

I like this because, often times, the person who says "Squeaky Wheels get the grease" are usually whining just to get what they want. Which, to me, is inappropriate.

The phrase "Empty Wagons make the most noise" is a way of saying they have no brains and aren't creative enough to find another solution other than whining all the time.

Well, as true as that may be, sometimes Silence is Golden¹:

2. Often the best choice is to say nothing.
   
   1897, Horatio Alger, Walter Sherwood's Probation, ch. 2:
   
   "But I have spoken long enough. There are times when silence is golden, and one of those times is at hand."

Both grease and gold have their uses, but more people would rather have gold than most other things if given the choice, and probably most notably including gunky grease. This is why things are often described as being worth their weight in gold¹, which is defined as "(idiomatic) Very valuable". Hopefully for what the words lack in literal weight, the wisdom of them up for by weighing greatly upon the mind.

I am tempted to explain in more detail, but going on too long would really run the risk of hypocrisy, per the following example from The Proverbs of Chaucer with Illustrations from Other Sources as found in Scotish Notes and Quiries, volume 6 no. 10 (march 1893):

The preacher has said that there is "a time to keep silence and a time to speak," and this counsel has passed into many a proverb. Speech in season, and a discrete silence when necessary are virtues which all commend but few find easy to practise. It has been said that Carlyle has taught us in thirty-seven volumes that Silence is golden. None has preached the value of silence more eloquently; none ever found it harder to put into practice what he preached. Let us look at a few of the proverbs bearing on this subject as we find them in Chaucer:

For this reason I am going to hold my tongue and let the worth of the phrase prove itself.

¹ Definitions referenced excerpted from Wiktionary, which licenses its text under CC-BY-SA 3.0 terms.

I don't believe this is a common phrase, but I like to use "Squeaky mouse gets the cat" because of the identical beginning.

You catch more flies with honey than vinegar.

As Wiktionary defines this saying:

It is easier to persuade others with polite requests and a positive attitude rather than with rude demands and negativity.

My response is always 'If the grease doesn't work then the wheel gets replaced'.

Retort it with

empty vessels make the most sound

Foolish, unwise, or stupid people are the most talkative.

idioms.freedictionary.com

More usually told to children than adults: " 'I want' never gets."

Roughly meaning that those who repeatedly say "I want this... , I want that ..." will be ignored in favour of more polite people.

I remember seeing a demotivational poster online long ago that fits, it went:

"The Tallest Blade of Grass is the First To Get Cut"

Where I grew up, people said "those who ask don't get". But I don't know if that's used anywhere else.

There's a saying in my tongue: the baby cries when the mother takes the breast away.

It's an interesting counter. If "squeaky wheel gets the grease" reflects the thought that complaining is the way to get what you want, most other answers aim to counter by claiming complaining will not get you what you want or worse. My answer says the complaining is an effect, not a cause, and you're being instructed on how to grow up and behave.

The correct phrase, a common English phrase which is the opposite of the one you gave:

"Closed mouths don't get fed."

Your statement says that a person who expresses their need or complaint has a better chance of getting their need met than someone who doesn't. This statement here says that a person who does not express their need or complaint will not get their need met.

In Australia we often talk about the "tall poppy syndrome". Here we are regarded as having an egalitarian outlook and tend to want to pull down those that either reached higher status (possibly illegitmately) or being too prominent and outspoken. So when someone gets remonstrated for stepping out of line, or sticking their neck out (yet more metaphors) they might declare they have been a "tall poppy" cut down by the community.

https://en.wikipedia.org/wiki/Tall_poppy_syndrome