A's and B's

Find all 5-letter words consisting solely of the letters A and B, such that there are no words with three repeated characters in a row (no AAAxx, xAAAx, or xxAAA words.) There are 16 such words. Make a 5x5 grid of A's and B's, such that each row (read left to right), column (read top to bottom), and the two diagonals (read top to bottom) is a different one of these words. Follow-up: how many such squares are there? Is it possible to make a square in which all the words beginning with "A" can be found?

Source: Original. Based upon Macalester College Problem of the Week #965, which asked the 4-letter version of this problem, of which there is a single solution.

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