Use a Mac App Store free app without having to associate/login to Apple ID?

An Apple ID in OS X is attached to the account, not to the machine so you can create a new account, use your Apple ID, use the app, then delete the account when finished.

Another alternative is to use another Apple ID. There is nothing stating that you can't have more than one Apple ID. Just create one for when you are doing tech work like what you are doing now and use that one.

Finally, you can ask your friend to sign in and use his account so you can download the app under his profile.

Since the application is free and doesn't contain any in-app purchases I would suggest you to email the developer that you would like to have a directly link to download the application. In most cases those people are generous and don't hesitate to provide you with an alternative. I am able to tell this because once I had some kind of problem with my AppStore and at the same I had to download Memory Clean. So I mailed the developers for a link and got an answer within 24 hours. Let me know if this worked :)