We can only say "sneezes into the handkerchief". This particular preposition applies to the noun "handkerchief".
If I could use a bit of a disgusting example:
If you sneeze into a cup, you are sneezing inside it, where the drink is.
If you sneeze onto a cup, you are sneezing on the outside of it, but not into it.
I'm not sure if this helps, but ju