Both iOS 5+ and OS X v10.7+ include the font Apple Color Emoji, and that font does have a glyph for U+2705 (the same glyph you see in my screen shot)², so OS X and iOS text rendering systems use it. On your Windows computer, I’m guessing the font that was substituted for Verdana/Geneva there was Segoe UI Symbol³.
As an aside, I’m pretty interested on how OS X and Windows decide on which font to use when there are multiple fonts installed containing the required glyph. For example, I have two other fonts on my OS X system that have a glyph for U+2705 (Everson Mono and Symbola), but OS X always seems to consistently pick Apple Color Emoji’s glyph. Maybe OS X’s text rendering system goes through the fonts in alphabetical order and uses the first one it finds containing the required glyph? It would be great if end users could have a bit more control over the font substitution process. I know it’s possible to do in some text editors like Emacs⁴, but I believe that programs like that use their own text rendering systems instead of that supplied by the OS (could be wrong though).
EDIT: Oh, I forgot to add, as for having the glyph be a green box with a white inner check mark, The Unicode Consortium actually mentions how the glyphs should look for characters with BLACK or WHITE in the name⁵:
“Q: What about characters whose names include WHITE or BLACK?
A: Names of symbols such as BLACK MEDIUM SQUARE or WHITE MEDIUM SQUARE are not meant to indicate that the corresponding character must be presented in black or white, respectively; rather, the use of “black” and “white” in the names is generally just to contrast filled versus outline shapes, or a darker color fill versus a lighter color fill. Similarly, in other symbols such as the hands U+261A BLACK LEFT POINTING INDEX and U+261C WHITE LEFT POINTING INDEX, the words “white” and “black” also refer to outlined versus filled, and do not indicate skin color.”
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¹ — https://en.wikipedia.org/wiki/Font_substitution
² — http://apps.timwhitlock.info/emoji/tables/unicode#block-2-di...
³ — http://www.fileformat.info/info/unicode/char/2705/fontsuppor...
⁴ — http://stackoverflow.com/questions/6491202/overriding-emacs-...