If you let other users execute a program, then they can know everything the program is doing, whether the program file is readable or not. All they need to do is point a debugger (or debugger-like program such as
strace). A binary executable can run if it's executable and not readable (a script can't, because the interpreter needs to be able to read the script), but this doesn't give you any security.
If you want others to be able to execute your program as a black box, without letting them see exactly what the program is doing, you need to give your script elevated privileges: make it setuid to your user. Only root can use debugging tools on setuid programs. Note that writing secure setuid programs isn't easy, and most languages aren't suitable; see Allow setuid on shell scripts for more explanations. If you're going to write a setuid program, I strongly recommend Perl, which has a mode (taint mode) that's explicitly intended to make secure setuid scripts possible.