It's a nice question. This Boolean algebra, known as the Lindenbaum algebra, is a countable atomless Boolean algebra — it is atomless because we can always take the conjunction of any formula with a new atom or its negation — and all such Boolean algebras are isomorphic by a standard back-and-forth argument. So your particular presentation of the algebra is inessential, since there are many presentations of this algebra.
The back-and-forth argument also shows that the structure is homogeneous — any finite partial isomorphism will extend to an automorphism of the whole algebra. Any finitely many parameters are contained in a finite Boolean subalgebra, and so any two elements that stand in the same relation to the atoms of that subalebra will be automorphic images. This severely limits the definable sets and will support a classification. Every definable set from parameters will be a finite union of the sets of points that all relate to the atoms of the finite subalgebra generated by those parameters in the same way. (Perhaps someone can provide a fully detailed account.)
In particular, the set of propositional atoms will not be definable with parameters, since any atom $p$ not appearing in the expressions of the parameters will be automorphic with the conjunction of two other such atoms $q\wedge r$ over those parameters. Similarly, no infinite set of atoms will be definable with parameters, since we can always move one of them automorphically to something that isn't an atom.