Optionally, move all your constants to .cpp file without declaration in .h file. Use anonymous namespace to make them invisible beyond the cpp module.
// MyClass.cpp
#include "MyClass.h"
// anonymous namespace
namespace
{
string RE_ANY = "([^\\n]*)";
string RE_ANY_RELUCTANT = "([^\\n]*?)";
}
// member function (static or not)
bool MyClass::foo()
{
// logic that uses constants
return RE_ANY_RELUCTANT.size() > 0;
}