SKNN regression problem
My best guess here is that your learning rate is way too high for the problem. You also probably have far more neurons in your hidden network than you need, seeing as you're using just one feature. 

Recall that learning rate is controlling the "step size" in gradient descent and that for your dataset, it is likely far too high. I made some minor changes to your code and got better results than linear regression. Notice the use of 2 hidden neurons, a 0.001 learning rate, and 20 iterations.

# Now using the sknn regressor
# http://scikit-neuralnetwork.readthedocs.org/en/latest/guide_beginners.html


from sknn.mlp import Regressor, Layer

nn = Regressor(
    layers=[
        Layer("Rectifier", units=2),
        Layer("Linear")],
    learning_rate=0.001,
    n_iter=20)

nn.fit(diabetes_X_train, diabetes_y_train)
print("Results of SKNN Regression....")


# The coefficients
print('Coefficients: ', regr.coef_)
# The mean square error
print("Residual sum of squares: %.2f"
      % np.mean((nn.predict(diabetes_X_test) - diabetes_y_test) ** 2))
# Explained variance score: 1 is perfect prediction
print('Variance score: %.2f' % nn.score(diabetes_X_test, diabetes_y_test))

# Plot outputs
plt.scatter(diabetes_X_test, diabetes_y_test,  color='black')
plt.plot(diabetes_X_test, nn.predict(diabetes_X_test), color='blue',
         linewidth=3)

plt.xticks(())
plt.yticks(())

plt.show()


SKNN regression: 

Results of SKNN Regression....
Coefficients:  [ 938.23786125]
Residual sum of squares: 6123.67
Variance score: 0.50