Which one is OK?
I'm not doing this because of what anyone would think.
I'm not doing this because of what anyone might think.

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If you mean that your actions are not influenced by the opinions of others  - I am not doing this because of what anyone else thinks.

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Both of them are fine

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The first one is not natural.  I agree with Hilary.  The best way to express it is “I am not doing this because of what anyone else thinks.”

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the second one is better I think)