The limits you conjecture are natural, and they are currently open. I believe the best known result is by Ping Ngai Chung and Shiyu Li, who proved that $$ \liminf_{n \to \infty} \frac{|E_{n} \cap \{1, 2, \ldots, n\}|}{n} $$ and $$ \liminf_{n \to \infty} \frac{|O_{n} \cap \{1, 2, \ldots, n\}|}{n} $$ are both $\geq \frac{1}{64}$. They can improve this to $\frac{1}{8}$ if they assume the Hardy-Littlewood prime $k$-tuples conjecture. The main tool used is Selberg's sieve and their paper appears in Integers in 2013. (See http://www.integers-ejcnt.org/vol13.html - the paper is A79.) They mention their result was obtained almost simultaneously by M. Alboiu.

The limits you conjecture are natural, and they are currently open. I believe the best known result is by Ping Ngai Chung and Shiyu Li, who proved that $$ \liminf_{n \to \infty} \frac{|E_{n}|}{n} $$ and $$ \liminf_{n \to \infty} \frac{|O_{n}|}{n} $$ are both $\geq \frac{1}{64}$. They can improve this to $\frac{1}{8}$ if they assume the Hardy-Littlewood prime $k$-tuples conjecture. The main tool used is Selberg's sieve and their paper appears in Integers in 2013. (See http://www.integers-ejcnt.org/vol13.html - the paper is A79.) They mention their result was obtained almost simultaneously by M. Alboiu.