Are both sentences OK?
It's not any of my business.
If I say,
It wouldn't be right for me talking about it.
It wouldn't be right for me to tallk about it.

And is this correct?
No one would want to be in his place / in his shoes right now.

Question

Are both sentences OK?
It's not any of my business.
  If I say,
  It wouldn't be right for me talking about it.
  It wouldn't be right for me to tallk about it.

And is this correct?
No one would want to be in his place / in his shoes right now.

Answer

"It wouldn't be right for me to talk about it." is correct.

Both the options for the second sentences are correct.