You first need to answer the question: "What type of correlation between these 2 categorical variables do I care about?"
Even for a 2 x 2 table like yours, there are at least 21 different objective measures of association to choose among, enumerated by Tan et al, Information Systems 29: 293-313, 2004. The choice among them has to do with what aspects of the association you wish to emphasize. The measures differ in terms of symmetry in variable permutation, invariance upon scaling rows and columns, and so forth.
Your two examples help to illustrate the issues. The phi coefficient recommended in a comment is closely related to the $\chi^2$ test; the $\chi^2$ statistic is the square of phi times the total number of counts ($N$) in a 2 x 2 table. That takes into account all observations in the 4 cells of the table, in a symmetric way. McNemar's test ignores the cases in which both are positive or both are negative, and uses only the cases in which the two classifications disagree to form its statistic.
The phi coefficient and its associated $\chi^2$ test might be the simplest to understand or explain to someone else, and it has the advantage of being calculated like the Pearson correlation coefficient in a 2 x 2 table.* But before you jump to use it, spend a bit of time thinking about the types of correlations you care about between your dichotomized measures.
Finally, having said all that, I still think that you will be better off examining CRP and IL6 as continuous values. Plots of one versus the other, perhaps overlaid with a smoothed curve as provided by loess, should provide a good deal more insight into the relationships between their values than will any categorical analysis based on arbitrary (even if "established") cutoff values.
*I've based this on a definition of phi that allows it to range over [-1,1] for a 2 x 2 table. Some define phi $= \sqrt{\chi^2/N}$, in which case it ranges over [0,1] and is equivalent to Cramer's V for a 2 x 2 table.
