By default, the Mac OS X identifies files without file extension as Unix Executable File. But often these can also be found as simple text files with instructions, version history, frequently asked questions, settings etc. and called like README, READ-ME, INFO, SETTINGS, OPTIONS etc.
Mac OS X tries to open files without extension with the Terminal app, which often causes an error message that the file doesn't contain any executable data.
If you want to change default application with Finder → Get Info option, you will only change the app for the current file. If you click on Change All... button you will get an error message.
However, there is a way how you can change default app for all files with no file extension found on your Mac.
Use RCDefaultApp to change default software for files without file extension
RCDefaultApp is free file association management tool for Mac that is installed as Preference Pane to System Preferences.
Default Apps main window
Download RCDefaultApp → Open the DMG archive → double click on RCDefaultApp.prefPane file.
Open System Preferences → Section Other → Click on Default Apps icon.
Mac System Preferences
Click on UTIs (Uniform Type Identifiers) tab.
Uniform Type Identifiers are used by OS X to identify common system objects, like documents, image file types, folders etc.
Scroll down the UTIs list and find public.unix-executable key and change Default Application to for example TextEdit.app.
Default Apps UTIs settings
Now files with no file extension will be opened in the TextEdit as a simple text files.
This solution changes the default app also for Unix Executable File, or other file types without file extension that will caused unexpected problems in system.
If you want to change an icon for files without file extension. Right click on a file → Get Info.
Drag a new icon archive with *.icns file extension by a mouse and drop it on default icon in upper left corner in Get Info panel.
Change default icon
If you want to set the default icon. Right click on file → Get Info. Click on icon in upper left corner and press the backspace key.