Will taking oxycontin and oxycodone cause me to fail a hair follicle test?

Yes: if they are testing for opioids (which they almost always do) AND you do not have a valid prescription for the medications. Even with a valid prescription, if you are in a safety-sensitive job, it still may be forbidden (like a commercial driver, dealing with dangerous machinery or even, in many cases, working as a health professional or in public safety - fire fighters and police).