Maria,
I do have the definitive response for regarding both these fonts.
First of all, they are publicly downloadable fonts on dafont.com. I downloaded them and examined them with a professional font editor and here is what I found.
For font JamesFajardo, the typographical opening and closing single and double quotes are actually defined in the font as blanks. As such, since the characters are defined, InDesign (and any Adobe application) would use them as-is. That results in the blank being displayed. If you want an apostrophe, keying that to replace the closing single quote will make your document displayable and printable.
For font Hanshand, the typographical opening and closing single and double quotes are not at all defined in the font. Thus, the .notdef glyph (that square box) is used. This is completely in compliance with the specification for use of these fonts. Like the case of the JamesFajardo font, if you want an apostrophe, keying that to replace the closing single quote will make your document displayable and printable.
By default, InDesign uses typographical quotes when keying or even placing or pasting text. Thus, if your Microsoft Word document had used a non-typographical apostrophe, then importing that would yield a conversion to yield what you see. Also, when keying in Microsoft Word (or other Office applications), if you have that option turned off, you won't get any attempt to do automatic substitution.
Bottom line though, is that both these fonts are basically flawed in that common characters used in Western Latin language layout are missing and in the case of JamesFajardo, the font is really defective in that it defines those characters as blanks.
For your entertainment and edification, I have attached to this response the output font tables for both those fonts so you can see exactly what is defined in them. Ironically, a similar chart appears on the dafont.com web pages associated with each of these fonts.
Sorry for the inconvenience you have suffered through, but there isn't much at Adobe we can do to further assist you with those fonts. 
- Dov