ButtonMinHeight is an option for ButtonBox which specifies the minimum total height in units of font size that should be allowed for the button.
The default setting ButtonMinHeight->1 forces a button to have a total height which at least accommodates all the characters in the current font.
ButtonMinHeight->0 reduces the total height of a button as much as possible, allowing buttons containing characters such as x and X to be different heights.
See The Mathematica Book: Section 2.8.11.
See also: ButtonMargins, ButtonExpandable, RowMinHeight.
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