Hypothetical situation: A druid has been captured by evildoers, who slapped a metal breastplate onto the Druid, causing a violation of his no-metal-armor rule and thus render him unable to cast spells, use supernatural or spell-like abilities for 24 hours. This particular Druid has four levels in Planar Shepherd, granting him Plane Shift (to one particular plane) as a spell-like ability. But since this is granted by a Prestige Class, rather than levels in Druid, is this ability blocked by the Druid's rule on no metal armor?
I'd argue the answer is no, Plane Shift is not affected and is still usable.
A druid who wears prohibited armor or carries a prohibited shield is unable to cast druid spells or use any of her supernatural or spell-like class abilities while doing so and for 24 hours thereafter.
The wording says Druid spells and class abilities specifically. Planar Shepard is a different class, and the ability comes from there. So you can still use it.
You also still meet the prerequisites for Planar Shepard, as you still have Wild Shape, even if you're not able to use it at that time. (See this answer for discussion on prerequisites.)
Note that you do still lose your spellcasting. Planar Shepard doesn't give you new spellcasting, it advances your Druid spellcasting:
At each level, you gain new spells per day and an increase in caster level (and spells known, if applicable) as if you had also gained a level in a divine spellcasting class to which you belonged before adding the prestige class level.
As that's still a Druid feature, you lose access to it due to the armor restriction.