Is it true that babies should only use baby walkers if they are wearing shoes?

No walker at all. I never recommend walkers at all.About 1/3 of kids in them will suffer a fall accompanied by a head injury.They may offer an easy source of entertainment and or ease baby monitoring, but a playpen is safer and doesn't increase injury risk.
No. It is safer to have shoes on so no injury occurs if they fall or hit their foot on anything.