I don't think there is an option in grep itself to fiddle with the semantics of the -w option. Some locales probably have slightly different definition of what constitutes a "word" but relying on that seems brittle at best. Perhaps it's simplest to explicitly specify what characters are allowed as word separators. Something like this, maybe?
egrep -c '(^|[ ])56677([ ]|$)' filename
The stuff between the square brackets are a space and a tab. (In some shells you need to type something like ctrl-v tab to enter a literal tab character.) Add more characters if you want other characters to be allowed as word separators. The regular expression means "beginning of line, or one of the characters between the square brackets", followed by your string (in this case 56677), followed by "one of the characters between the square brackets, or end of line". Plain grep does not understand this syntax; it's "extended regular expressions", hence, "extended grep" aka egrep. (Though POSIX specifies a way to use a similar set of operators with plain grep, too, I believe).
Note also that you don't need or want the cat there; grep can read what input files you want to feed it all by itself.
The second example is the more intuitive way of thinking through the process: display the file and then specify what is being selected. The first is the more 'perfect' unix programming solution since grep does not need a cat command - in english, select for something taking a file as input.