This likely has been thought of before, but just in case it hasn't, here goes:

To prevent "missing character" box icons from showing up when OpenDyslexic is used, add the unicode-range font-descriptor to the @font-face declaration, limiting the font to only the characters in the range.

@font-face {
	font-family: 'opendyslexic';
	src: url('chrome-extension://__MSG_@@extension_id__/fonts/opendyslexic/OpenDyslexic-Bold.otf');
	font-weight: bold;
	font-style: normal;
        unicode-range: /* range of characters in OD */
}

@font-face supports the unicode-range font descriptor, which defines the range of characters for which a given face may be used. Adding unicode-range to the above with a value reflecting the available characters in openDislexic (OD) would mean a fallback font would be used for any characters not in the range.

Using Hebrew as an example (source: CSS: The definitive guide), the following would limit using CMM-Ahuvah to only the small range of Hebrew characters:

@font-face {
    font-family: "CMM-Ahuvah";
    src: url("cmm-ahuvah.otf" format("opentype");
    unicode-range: U+590-5FF;
}