Possible first steps in solving it
This is a partial answer, likely somewhere between 5% to 30% complete. But since nobody answered this yet, it might be a small breakthrough.
First, I just copied this from a comment to a chat link (thanks for Jerry Dean) and the OP confirmed that they are all correct:
Ok, are those random names?
The possible starting squares are Leona Lewis, Lennox Lewis, Lucy Liu and Lindsay Lohan. They all have the initials LL.
Gilbert O'Sullivan, Gary Oldman, Georgio O'Keeffe and George Osbourne. All of them have the initials GO.
Also there are a lot of people with G, L, W, Y and O as initials. This is improbable if the 27 people with known names were chosen at random or if their actual names didn't matter and they were chosen only by some other criteria.
If this phenomenon is not a red herring intended to mislead people, this should be some sort of word puzzle.
Also, considering this comment form the OP:
@msh210 Well, I haven't actually explicitly said that every space must be visited (just that you need to make sense of the selections...) - but if you think they do need to be then I'd hope any answer would justify why :)
This means that:
Not every square must be necessarily visited.
Some of the names might be randomly picked after all, and those would possibly be just red herrings. However, unvisited squares might be useful somehow, so I'm not sure.
Also, connecting the names, I produced this:
Red connections are people who share both initials.
Blue connections share only one initial.
Based on that, A should be one of:
George Washington, Abraham Lincoln, Ulysses Grant, James Garfield, Grover Cleveland, Dwight Eisenhower, Lyndon Johnson, Richard Nixon, Gerald Ford, George H. W. Bush, George W. Bush
And B should be one of:
George Washington, William Harrison, Rutherford Hayes, Abraham Lincoln, William McKinley, Theodore Roosevelt, William Taft, Woodrow Wilson, Warren Harding, Franklin Roosevelt, Dwight Eisenhower, Lyndon Johnson, Richard Nixon, Ronald Reagan, George H. W. Bush, George W. Bush, Donald Trump
Now, I got stuck.