It depends who you ask.
To many folks who answered the ELU question Sentence Construction: “Just Because … Does Not Mean”, it seems it is ungrammatical.
However see also this response to the contrary under Can a “because clause” be a subject clause? [duplicate]. See also the answer by snailboat that cites the CGEL.
According to explanations found in many traditional grammars, the because clause forms a subordinate clause and cannot be used as a subject. According to this line of thinking, this is just as ungrammatical as the sentence without just:
Because people don't talk doesn't mean they don't think.
Frankly, the version without just sounds "off" (questionable) to me. For those who find this ungrammatical, they might point out that using a different conjunction is just as ungrammatical:
Whenever people don't talk doesn't mean they don't think.
Therefore, some would say that the grammatical version is:
Just because people don't talk, it (or: this/that) doesn't mean they don't think.
However, as noted, the original version is certainly used in vernacular or everyday speech.
Now we come to the power of the people. If the people want to use "because" in the same way they use that, then the sentence can be judged as grammatical--although this may not be accepted by everybody.
As for the that clause, it is grammatical to say/write:
That people don't talk doesn't mean they don't think.
This is grammatical, (subordinate 'that' clauses can function as the subject) but people don't go around talking like this. Perhaps this is one reason that the specific construction you ask about is popular.
However, you may find it practical to appease everybody by not using the (just) because versions in writing and formal/business speech.