I've been through the PHB and DMG and not been able to find any specific rules for this. Clearly the string instruments must be played two-handed, but I don't see why my bard couldn't be playing the pan flute or horn in one hand (as a focus for his bardic spells) and wielding a weapon or shield in his other.
Pan flutes vary in size, as a simple Google image search reveals. Depending on the style, it might require one or two hands. A simple 4 or 5 pipe flute is certainly designed for single hand use, while a 20 pipe flute may not be reasonable to use in a single hand. In extreme circumstances, you might even argue it occupies a shield slot.
If you can play with no hands, you should be able to play with one hand.