What's the most appropriate statistical analysis to run for a correlational study?
Although you have two times of testing, you really just have two binary variables and you'd like to know if they are associated, there are no repeated measures here. You'd like to know if the value of one binary variable depends on the value of the other binary variable. I'll call them A,B and X,Y.
If there is no relationship between A,B and X,Y, then you expect the ratio of X:Y to be the same for A as it is for B (this is your null hypothesis). You can examine your data by building a 2x2 contingency table with cells AX, AY, BX, and BY.
You can test the null hypothesis described with a chi-square test for independence, or Fisher's exact test if the sample size is fairly small.