Opposite of "Squeaky wheel gets the grease"
Solution 1:
One proverb that came to mind is
The nail that sticks up gets hammered down.
Doing a web search, I found that apparently it has a Japanese origin, but I think it's common, or at the very least understandable, in English.
I also found an English.SE thread about the phrase, in which ps.w.g offers the phrase
The squeaky wheel gets replaced.
Solution 2:
I've always countered with "the quacking duck gets shot".
Solution 3:
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
Solution 4:
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.
Solution 5:
Well, as true as that may be, sometimes Silence is Golden1:
- 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.