Yes, bleed is negated
The rules for Damage Reduction say:
Whenever damage reduction completely negates the damage from an attack, it also negates most special effects that accompany the attack, such as injury poison, a monk’s stunning, and injury-based disease.
such as means those are examples, not an exhaustive list. This means that any ability that has a secondary effect that is tied to the attack will be negated if the damage is also negated (unless specifically said otherwise).
Bleed is a condition
It is common for bleed effects to mention that they ignore damage reduction (ie: Bleeding Attack and Bleeding Blow). However, the general rules for Bleed (neither here nor here) do not mention anything at all about damage reduction:
A creature that is taking bleed damage takes the listed amount of damage at the beginning of its turn. Bleeding can be stopped by a DC 15 Heal check or through the application of any spell that cures hit point damage (even if the bleed is ability damage). Some bleed effects cause ability damage or even ability drain. Bleed effects do not stack with each other unless they deal different kinds of damage. When two or more bleed effects deal the same kind of damage, take the worse effect. In this case, ability drain is worse than ability damage.
And it's not a type of damage listed under those that always ignore damage reduction either:
Damage Reduction does not negate touch attacks, energy damage dealt along with an attack, or energy drains. Nor does it affect poisons or diseases delivered by inhalation, ingestion, or contact.
The (optional) rules for Called Shots mention that the bleed damage caused by called shots is not reduced by damage reduction. So, it's clear in that context.
The eidolon's Bleed evolution doesn't make an exception and, apparently, should be affected by damage reduction as any other physical attack that inflicts the bleed condition.
An eidolon gains the ability to inflict bleeding wounds. Select one type of attack. Attacks of that type deal 1d6 points of bleed damage. This evolution can be selected more than once. Each time an eidolon selects this evolution, it applies to a different attack. The bleed effect doesn’t stack.
However, keep in mind that Bleed is a condition (that applies bleed damage at the start of the creature's turn), and damage reduction only helps against normal attacks (ie, with claws or manufactured weapons):
The numerical part of a creature’s damage reduction (or DR) is the amount of damage the creature ignores from normal attacks.
So, your eidolon's Bleed ability doesn't bypass damage reduction, and if the target has damage reduction, you have to first cause more damage with your attack to cause at least 1 point of damage. Then, that 1d6 bleed damage won't be reduced by their damage reduction each round, as bleed is a condition, not a normal attack.
For reference, the design intent seems to be that all bleed effects to bypass damage reduction, as mentioned by Jason Bulmahn (Developer) on this thread when answering about the Critical Hit Deck (back in 2007):
Bleed damage bypasses DR. Of course, in your game, you can rule it anyway you like. That was one of our design choices on this product. We give you the basics, but because the deck represents a fundamental shift in how crits work, we wanted to make sure it had the flexibility to work for your game.
Still.. if it was me, Bleed would bypass DR.
