Assuming that files with those names exist in multiple places in the directory tree and you want to preserve all of them:
find . -type f ! -regex ".*/\(textfile.txt\|backup.tar.gz\|script.php\|database.sql\|info.txt\)" -delete
Assuming that files with those names exist in multiple places in the directory tree and you want to preserve all of them:
find . -type f ! -regex ".*/\(textfile.txt\|backup.tar.gz\|script.php\|database.sql\|info.txt\)" -delete