The answer depends on the hypothesis/-es you want to test. If you simply want to know if activity at either of these time points correlates with reaction time, then you should compute two separate correlations. You could then make statements like "reaction time is significantly correlated with y1, but not with y2". To additionally say something about whether one correlation is significantly different from the other, you'd have to do a test on the difference between them.
With regards to testing this difference, it doesn't matter in this case if one of the correlations is not significantly different from 0. In fact this is a common mistake that people make when interpreting patterns of results, where they will assert that a is significantly different from b, based on the fact that a is significantly different from 0 while b is not. To actually be able to claim this, you have to test a against b directly, and it is perfectly possible for that test to fail to reject the null hypothesis under those circumstances.
I don't see for what reason concatenating y1 and y2 would make sense. If you want to know if y1 and y2 explain independent variance in X, then the thing to do would be to run a multiple regression analysis with y1 and y2 as predictors (and an intercept as the third predictor) in the regression model and X as the outcome variable, and compare that model to a model with only y1 or y2 (and again including an intercept).
If you want to know whether there is any relationship between y1 and y2, and X, you could test the model with both predictors against a model with only an intercept. Opinions vary (I believe) on whether it is better to start with a full model and then work your way down the various nested hypotheses, or start with the simple hypotheses and then build to more complex models.
To compare regression models, you would use an F-test. In general, I would advise you to read up on regression analyses. Note that a correlation is simply a normalized version of a regression, where the regression slope is expressed in terms of the variance shared between the predictor and the outcome variable. Multiple regression in turn is equivalent to a partial correlation analysis (where you assess whether each predictor variable explains unique variance after the effects of the other predictors have been removed).