What is the client language ? That is, we'd have to be sure exactly what datatype of username is and what the Replace method does in regard to that datatype. Also how the actual concatenation works for that datatype. There may be some character set translation that would translate some quote-like character in UTF-8 to a "regular" quote.
For the very simple example you show it should just work, but the performance will be awful (as per Thilo's comment). You'd need to look at the options for cursor_sharing
For this SQL
select * from user where username = '[blah]'
As long as [blah] didn't include a single quote, it should be interpreted as single CHAR value. If the string was more than 4000 bytes, it would raise an error and I'd be interested to see how that was handled. Similarly an empty string or one consisting solely of single quotes. Control characters (end-of-file, for example) might also give it some issues but that might depend on whether they can be entered at the front-end.
For a username, it would be legitimate to limit the characterset to alphanumerics, and possibly a limited set of punctuation (dot and underscore perhaps). So if you did take a character filtering approach, I'd prefer to see a whitelist of acceptable characters rather than blacklisting single quotes, control characters etc.
In summary, as a general approach it is poor security, and negatively impacts performance. In particular cases, it might (probably?) won't expose any vulnerabilities. But you'd want to do a LOT of testing to be sure it doesn't.