What is the correct name for a part of a chessboard: 'square' or 'cell'? I play shogi (Japanese chess) and there is a word "マス" to call a cell of a square lattice not only used in shogi and chess. As far as I have read the article about chess in Wikipedia, it seems that "square" is used more often than "cell" to call "マス" in chess. What is the most natural term to call a cell of the chess board? Is it "square"?
Yes, that's a "square". The word "cell" is never used. E.g. from the Wikipedia article:


  Chess is a two-player strategy board game played on a checkered board with 64 squares arranged in an 8×8 grid.





  The 64 squares alternate in color and are referred to as light and dark squares.


Or, if you prefer a dictionary:


  3 : any of the quadrilateral spaces marked out on a board for playing games


(source: Merriam-Webster)


  1.3 A small square area on the board used in a game.
  ‘move the white king's pawn forward two squares’


(source: Lexico, more example sentences about chess in that link)

Finally, as @chaosflaws notes in the comments, the official FIDE Laws of Chess:


  2.1 The chessboard is composed of an 8 x 8 grid of 64 equal squares alternately light (the ‘white’ squares) and dark (the ‘black’ squares).