Any subset of the collection of triples (a,b,c) with a,b,c all in {0,1} can easily be described using and, or, not: Make the corresponding truth table, and for each row write that row as a conjunction of "literals", i.e. v or (not v) for a variable v.
For example if the vars are x,y,z in that order, one row of a truth table would have x true, y false, and z true. Then this row corresponds to the conjunction
x and (not y) and z.
Now just put the rows that appear each in this format, put parentheses around each row result, and put "or" between them in case there are more than one row giving true. This is a standard method to get a boolean from the initial collection of triples.
I think you're asking whether one can realize any such collection of triples by means of a plane which cuts the cube without going through any vertices, and using one side of that plane for the "true" triples.
But this can't give all possible subsets, since the set {(0,0,0), (1,1,1)} would be two diagonally opposite points on the cube, and any plane cutting the cube with these two on one side will definitely have more triples on that same side. The boolean for this is of course (not x and not y and not z) or (x and y and z).