The HTTPS Everywhere FAQ does not make this claim:
HTTPS Everywhere protects you only when you are using encrypted portions of supported web sites. On a supported site, it will automatically activate HTTPS encryption for all known supported parts of the site (for some sites, this might be only a portion of the entire site). For example, if your web mail provider does not support HTTPS at all, HTTPS Everywhere can’t make your access to your web mail secure. Similarly, if a site allows HTTPS for text but not images, someone might be able to see which images your browser loads and guess what you’re accessing.