I'm not sure why you would. I don't think POSIX generally specifies the behavior of command line tools in such a level of detail. FWIW, the regex type used by default by GNU Grep is already POSIX's Basic Regular Expressions. (It also supports POSIX Extended Regular Expressions and PCRE2.)
Afaik, EulerOS and other Unix-certified Linux distros just ship the usual GNU userland.