Is it proper to have a hankerchief in the dress shirt pocket when not wearing a suit jacket?

Nope.
The handkerchief is for the jacket pocket only.
A proper dress shirt should not have any pockets.

No

I'll third that. It's a big no.

Although many shirts have front pockets, a handkerchief is only intended for a suit jacket, not the shirt. It's a big fashion no no.