If you're fine with house-ruling things, then you can do anything, so yeah you can do this.
Just using the character creation options presented in the books, no you can't do this.
I only have access to the hybrid rules in Player's Handbook 3, and as far as I'm aware there are no other generalized hybrid rules available.
Hybrid Classes are Different Classes
If you look at the Hybrid Characters section in Player's Handbook 3, you'll see that you're not combining the actual classes from the other books. You're combining the hybrid classes presented in this section.
This is most evidenced by the Class Traits blocks for each hybrid class, which do not match the Class Traits blocks for the normal classes. They have different class features, which have been balanced specifically for each class, rather than derived from the base classes in a mechanistic or algorithmic way.
What this means is there doesn't need to be a rule stating whether you can make hybrid characters with Essentials classes and Player's Handbook classes or not. You can't make hybrid characters with Player's Handbook classes and other Player's Handbook classes. You make hybrid characters with hybrid classes, so in order to make a hybrid character using an Essentials class, there would need to be a corresponding hybrid class.
Brian Ballsun-Stanton has pointed out that some additional hybrid classes have been introduced in Dragon Magazine, including ones corresponding to some Essentials classes in DM 400 and 402. Specifically the Druid (Sentinel), Warlock (Binder), Paladin (Cavalier), Paladin (Blackguard), and Assassin (Executioner).
Lucky for you, the class you want to use, Blackguard, is included in this list. If you wanted to use another class, like maybe Fighter (Slayer), then you would have to houserule it.
Hybrid Classes Make Assumptions
The rules for hybrid classes make a lot of assumptions based on the rigid way in which the classes in the Player's Handbooks are defined. These assumptions are not necessarily true for Essentials classes, but are fundamental to balancing the hybrid characters.
For instance, every Player's Handbook class gets an Encounter power at level 3, except for classes with psionic augmentation, and there's a large, complicated section on how to handle the psionic classes. It's this assumption about getting encounter, daily, and utility powers at an equal rate from each of the two classes that is used to balance how much you can take from each class, and also to ensure that a minimal level of functionality is taken from each class. A blackguard gets an extra use of Dread Smite at level 3, not a new encounter power. How would the hybrid rules handle this?
All that being said, hybrid characters aren't super well balanced anyway, so whatever.
You'll have to look over the existing hybrid characters to get a feel for the kind of balance decisions involved, then apply those ethics to the classes your player wants to hybridize. This will be easier the closer in structure the class you're dealing with is to the Player's Handbook classes. It should also help to look at the hybrid versions of the Essentials classes introduced in Dragon Magazine, if you have access to that.
Keep in mind the basic idea that you should have access to options from both classes, but not be able to stack constant benefits from both classes at the same time for the same thing. For example, a hybrid rogue/ranger can only use Hunter's Quarry damage dice with Ranger powers, and can only use Sneak Attack damage dice with Rogue powers, so he can't apply both Sneak Attack and Hunter's Quarry damage to the same attack.
Also keep in mind the general idea that, at each level, you can choose which class to get a benefit from. This may not be as clean with the Essentials classes, but you can still do it if you're careful.
Although there is apparently a hybrid Blackguard out there to use, I'll still use the Blackguard as an example of the types of problems you might run into when attempting to adapt another class.
For example, your hybrid blackguard at level 3 could choose to get either an additional use of Dread Smite (essentially a new encounter power that just so happens to be identical to an already known encounter power) or a level 3 Warlock encounter power. However, the hybrid rules require to to pick an encounter from one class at level 1, and an encounter from the other class at level 3. If you picked Dread Smite at level 1, then you have to get an encounter from Warlock at 3. If you picked an encounter from Warlock at 1, then you would get an additional use of... wait a minute I don't know Dread Smite how do I get an additional use of it? This particular mismatch is simple to resolve, just learn Dread Smite for the first time at level 3, but others might require more thinking.