Do Engle-Granger residuals need to be normally distributed? Do EngleGranger residuals need to be normally distributed?
 A: The Engle and Granger test for cointegration is basically an ADF test in the estimated error of the long run equation, say $\hat{u}$. The ADF test is looking for stationarity of this process (or trend-stationary). It does not require normality, as there are many non-normal processes which are stationary (see here or here).
For instance, this paper compares the power of different unit root tests (including ADF) for a whole range of distribution of the variable in question. Distributions include normal and non-normal like Chi-squared, Beta, and Laplace (double exponential in their terminology). Their interest is to propose more powerful unit root tests, but an aside conclusion is that, even under non-normality, the ADF test performs "well" (their proposed tests perform better though). 
Having this in mind, the next question is: does the error in a cointegrated, long run equation has to be normal? This is akin to the question of whether errors in a regression have to be normal. There are tons of related posts in CV about this. The short answer seems to be: hopefully yes, but might not be. Examples here, here, here, and more generally, here.
(Richard Hardy's summary of this normality issue in the comments is certainly more precise on this than my summary. Basically, if you have a small sample, you want normality, so that standard tests have the theoretical distribution the should have (so, tests are valid). If you have a large sample, the Central Limit theorem will do that job for you.) 
